Probability (or Randomness): Ontic, Epistemic, Aleatory, Subjective, Personalist, Bayesian, Egocentric, Quantum, Objective, Rational, Inductive, Existential… ?

Estimated read time (minus contemplative pauses): 118 min.

1. Introduction

In his 2009 paper, “On ‘Why’ and ‘What’ of Randomness,”1 Soubhik Chakraborty, among other things, “argues why there should be four motivating factors for calling a phenomenon random: ontic, epistemic, pseudo and telescopic , the first two depicting ‘genuine’ randomness and the last two ‘false’”2. The paper overall, writes Chakraborty, “tells (the teacher) not to jump to probability without explaining randomness properly first and similarly advises the students to read (and understand) randomness minutely before taking on probability (Chakraborty 1, Abstract).

This is not an essay, but rather is a collection of notes I made while reading the paper, semi-cleaned up for others—Hi future self! :)—to reference. I don’t aim to approach the paper critically, but rather as a guide for getting a better handle on some of the concepts it covers, both as Chakraborty understands them and as they are understood in the related literature more broadly. In fact, the latter is what the bulk of these notes are dedicated to.

Chakraborty’s injunction to understand randomness before approaching probability—a distinction he strongly emphasizes—seems sensible, even if one isn’t dealing with probability in a context of computer science, a field in which pseudo and telescope randomness seem particularly relevant. I won’t deal here with those depictions of “false randomness” (as he puts it). I’ll focus, rather, on the “genuinely random” ontic and epistemic randomness. Like it or not, my attempts to understand Chakraborty’s of these concepts will be filtered through my years of thinking about ontic and epistemic probability. So, part of my aim here will be to tweeze apart the probabilistic and randomness-oriented applications of these terms, even if it means working backwards in order to turn around and face forwards (given Chakraborty’s aforementioned injunction). If I can clean up—improve the consistency of—my own conceptual and semantic framework for talking about kinds of randomness and probability, that’d be nice. A more realistic goal (and the one I end up getting closer to) is to sharpen my respect for the difficulty in doing so.

Chakraborty, who repeatedly emphasizes that his focus is on randomness and not probability, directs us to Joseph Hanna for a discussion of ontic probability, and to Ian Hacking for one of epistemic probability. I’ll say more about these and several other sources in a moment—in fact, that’s where the bulk of our time here will be spent.

First, some preliminary comments about the concepts in question.

2. Ontic and Epistemic Probability: A Preliminary Look

By “preliminary,” I mean my already existing understanding of these terms, which I’ll review off the top of my head before looking at their counterparts under Chakraborty’s and others’ usage.

2.1 Epistemic Probability

Simply put, I take epistemic probability to involve the assignment of probabilities according to what is known. This may or may not be the “real” or “true” probability of the event in question. For example, a newly purchased die is likely not perfectly fair, but we assume it is given what we know: looks symmetrical, is packaged as a fair die, doesn’t feel weighted, dice that look and feel like this one are known to land on a given side with a certain relative frequency, and so on. And so we assign a probability of 1/6 to the die landing on any particular side, even though this may not be the true probability.

We might think we can take the notion of epistemic probability even further here by observing that a die roll is causally determined, so that if we knew enough about the pressure with which the die is rolled and its starting position and its immediate environment, etc., we could assign a probability of 1 to the die landing on a particular side. But we lack that knowledge, so we say, on usual grounds, 1/6 for the die’s unknown outcome, both before and after rolling it (though the latter could alternatively be characterized as the probability that you guess correctly).

I refer to “the usual grounds” for assigning 1/6. There are debates to be had about that. As James Franklin puts it in his 2001 (with a new Preface added in 2015) book, The Science of Conjecture: Evidence and Probability before Pascal:

The debates of the time of Laplace on the principle of insufficient reason never satisfactorily resolved whether the probability of a coin’s landing heads is a half because the coin is symmetrical, and hence there is no reason to prefer heads to tails, or because many throws of coins have been observed to produce about half heads and half tails. Keynes’ chapter on the weight of evidence shows that we are still no closer to explaining the difference between the probability of a hypothesis in the two cases in which there is little evidence with a certain probability on balance and in which there is a great deal of evidence with the same probability on balance.3 Are there two dimensions of probability, one giving the total probability and the other the firmness or weight with which that probability can be held? (Franklin 77–78).

The above gives an indication of the difficulty of determining what sort of knowledge informs the assigning of probabilities, even when those assignments are uncontroversial. But wherever one lands in the above debate, one’s assignment comes down to a weighing of evidence, which is another way of referring to what one knows about the event in question.

There is a certain sense in which all probability is epistemic. The least interesting example of which would be the probability you assign when you know everything worth knowing about an event and you know you know this, and you know this is getting you to the best possible probability assignment. I will draw a harder line here than this, and will say that epistemic probability carries with it the idea that you at the very least don’t know with certainty whether you know everything worth knowing, and, even more often, you are very open to the possibility that, with more information, you would revise your probability. If you are assigning the perfectly correct probability, it as it least to some degree a matter of luck.

In short, on this strict view, epistemic probability is what you assign when you’re doing the best you can with what you know, which may range from a little to a lot, and you can’t be sure how close “a lot” is to “everything.”

There will be plenty of opportunity below to further complicate the concept of epistemic probability, not to mention other concepts I’ve evoked here, like that of “true” probability.

2.2 Ontic Probability

I understand ontic probability to involve events that are inherently probabilistic, which is to say that, even in principle, their outcomes can only be assigned probabilities between 0 and 1. We might also think of this as involving “true randomness,” or the idea that probabilities are baked into the natural world. Here, I’ll use “true randomness” and “ontic randomness” interchangeably, though we’ll see if that jibes with Chakraborty’s explanation. In fact, I generally use terms like “truly probabilistic” and “inherently probabilistic” and “ontically probablistic” interchangeably (and you can exchange the word “probabilistic” in those with “random”).

Here’s an illustration of the concept.

I mentioned above that, given enough information, we might be able to predict the outcome of a die roll with certainty. But if a perfectly fair die roll, vigorously executed with no cheating etc., is inherently probabilistic, which is to say truly or ontically random (as I currently understand those terms), then the best that can be done is an assignment of 1/6 for landing on any particular side. This means that 1/6 the best that even God can do. It is literally the correct, most perfect answer.

If the die turns out to be just shy of fair, God, in here supreme omnipotence, will assign the correct, most perfect answer of, say, 1/6.00000000000000001 for the die landing 3.4 God has thus predicted what will happen with certainty: it is certain that the die will land 3 with a probability of 1/6.00000000000000001. While we mere mortals will be epistemically stuck at 1/6 until we figure out a way to make this prediction accurately.

My view of ontic probability is a rather strict one. That is, there might be a temptation towards a softer view in which the complexity of a determined event makes it prohibitive to work out its true probability. Suppose I want the probability of how a fair die will land if we chuck it into a running air mix machine (think lottery ball selector) just as the machine is being whisked into the Pacific ocean on a small raft with the idea that as soon as we land on the other side the machine will launch the die onto the face of a craggy rock, down which the die will cascade until it comes to rest in the moist sand. We might say the complexity of such a thing is too much to compute, even in principle, and so the probability is ontic.

I’m going to draw a hard line and say that complexity is not enough to make a probability (or randomness) ontic. If the surrounding events are deterministic, then, in principal, God can predict precisely how the die will land with a probability of 1, even if such a thing is hopelessly out of reach for minds like ours and the computing power of the best computers minds like ours can imagine. If, however, the surrounding events are indeterministic in a literal sense (more about which below), then the events are inherently, or ontically, probabilistic, and the most perfectly accurate prediction possible is some number between 0 and 1.

There is also a temptation to take “ontic” to refer to whatever the probability is one should assign given some definition of probability—e.g., statistical frequency. So, the “true” probability of a perfectly fair die landing 4 is 1/6, because xyz (see again the above Franklin quote). This will prove more than once highly problematic position to take, however, except in the case (I claim) in which it is the probability God would assign.

(Though I would use the word “ontic” to mean the true nature of something in some contexts, such as: epistemic versus ontic reality, which is roughly analogous, I take it, to Kant’s phenomenal-noumenal distinction, and to something like the umwelt-veridical and subjective-objective reality distinction, etc.—for an example of epistemic-ontic language in this context, see this 2003 book chapter: “Epistemic and Ontic Quantum Realities” by Harald Atmanspacher and Hans Primas.5)

I’ll now have a look at how “epistemic,” “ontic,” and related terms are used in a handful of other sources.

3. Survey of Term Use

I could spend goodness knows how much time digging around for uses of these terms. Instead, I’ll pick some popular references and a few books I’m familiar with that deal with probability and randomness, and will conduct a brief survey of how they talk about the concepts in question. This also gives a sense of how my own conception of these terms has come about.

I’ll end the survey with a paper by Joseph Hanna that was unknown to me until I encountered Chakraborty’s mention of it.

Again, the goal here is more to learn or build a reference guide than to criticize or argue, but I’ll of course at times point out dissonances and consonances etc. where I perceive them, and will share my own opinions when appropriate.

3.1 A Note on Background Terminology

3.1.1 Epistemic

“Epistemic” is an adjectival form of “epistemology,” which is to say that it has to do with knowledge or belief. “Epistemic” is more or less synonymous with “epistemological,” but the former is more far more commonly used in the present context. “Epistemic probability” does often get exchanged, however, with the notion of subjectivity, as in the term, “subjective probability,” which is also often called “Bayesian probability.” More on all this soon.

In his paper’s Extended Abstract, Chakraborty defines “epistemic” along those same lines: “Epistemic is what we know (hence it relates to our knowledge of something).” I’ll reserve consideration of his actual use of the term for after the survey.

3.1.2 Ontic

“Ontic” and “ontological” are closely related terms, but are not generally interchangeable. If you dig around for uses of “ontic,” you’ll find that different people us it in different ways. Namely, when they’d like to home in on something about the nature of what’s real, while keeping separate the broader notion of ontology, which is a more commonly and consistently used word for dealing with concepts related to being.

I noted a moment ago that using the word “ontic” to mean just whatever the true probability of an event is, rather than the event being objectively probabilistic (or random), will prove problematic. Another reason I like “ontic” is, in fact, for its relation to the word “ontological.” That is, to say that an event is ontological probabilistic, or that randomness is part of the ontology of an event, is to say that those qualities are essential, defining features of the event. Much as we say that there is some permeating quality of treeness, without which a thing cannot be a tree. (Who knows what that is. One way to explore the question is to start removing things from your concept of tree until the concept is not of a tree. At what point does that happen? At the leaves? The bark? The roots?)

Anything we think of as a particular sort of thing has satisfied some ontological condition for being that sort of thing, as vague as such conditions may be. Events that are inherently probabilistic have randomness in their ontology much in the way a water molecule has hydrogen in its ontology. Arguments about the conditions for social group membership are arguments about social group ontology. And so on.

To be clear, though, this is just a question of which words to apply to phenomena, on the assumption that the phenomena—even if they in some cases arise from discursive practices, and even if essentially as sets of concepts—exist whatever we happen to call them. I’m personally interested in the project of making consistent use of terms like “ontic probability,” and am pleased that this endeavor happens to encourage deeper thinking about the denoted phenomena in themselves.

That said, I’ll assume the unremarkable definition of “ontic” as a word for talking about what’s real or what truly exists, however we choose to apply this to probability or randomness. This, again, aligns with the definition Chakraborty gives in his Extended Abstract: “Ontic in the article means what is actual (irrespective of our knowledge).”

(For a deeper look into the terms “ontic” and “ontological” than I’ll get into here, a good start might be the Wikipedia entries on Ontic and Ontology, as well as the Stanford Encyclopedia of Philosophy (SEP) entries on Heidegger, Logic & Ontology, and Social Ontology)

3.1.3 Related Terms

Yet other terms related to probability and randomness will come up here, including several that don’t show up in Chakraborty’s paper—e.g., aleatoric, personalist, inductive, etc. I’ll discuss these as I encounter them.

I think this a sufficient starting point for looking at instances of usage.

3.2 Wikipedia

Wikipedia is a collective effort that sometimes does a great job but, more importantly for our purposes, it also gives a sense of what some people thinking about this stuff take our terms to mean at the time of this writing. Not to mention that it’s likely to be among the first places inquiring minds go to for investing these concepts.

When block-quoting from entries, I may or may not include links and citations from the original text.

The Wikipedia entry on Probability makes brief mention of some of the concepts we’re interested in, but a much better place to look is the more philosophically motivated entry on Probability Interpretations, which features three relevant sections. The first of these is titled “Subjectivism” and includes the following characterization of epistemic probability as Bayesian/subjective:

Subjectivists, also known as Bayesians or followers of epistemic probability, give the notion of probability a subjective status by regarding it as a measure of the “degree of belief” of the individual assessing the uncertainty of a particular situation. …

The section goes on to give some basic examples and to note that Bayesians often point to the work of Frank Ramsey and Bruno de Finetti as justification for their position. This section also links to a full article on subjectivism, titled Bayesian Probability. The term “epistemic probability” doesn’t show up there (it favors the term “subjective probability”), but the article does mention “aleatoric and epistemic uncertainty” defined as “uncertainty resulting from lack of information,” and in fact links to a full article on that topic, called Uncertainty Quantification, in which there’s a section called “Aleatoric and Epistemic Uncertainty.” It’s worth it to reproduce the bulk of that section:

Uncertainty is sometimes classified into two categories, prominently seen in medical applications.6

Aleatoric uncertainty is also known as statistical uncertainty, and is representative of unknowns that differ each time we run the same experiment. For example, a single arrow shot with a mechanical bow that exactly duplicates each launch (the same acceleration, altitude, direction and final velocity) will not all impact the same point on the target due to random and complicated vibrations of the arrow shaft, the knowledge of which cannot be determined sufficiently to eliminate the resulting scatter of impact points. The argument here is obviously in the definition of “cannot.” Just because we cannot measure sufficiently with our currently available measurement devices does not preclude necessarily the existence of such information, which would move this uncertainty into the below category. The term “aleatoric” is derived from the Latin alea or dice, referring to a game of chance.

Epistemic uncertainty is also known as systematic uncertainty, and is due to things one could in principle know but doesn’t in practice. This may be because a measurement is not accurate, because the model neglects certain effects, or because particular data has been deliberately hidden. An example of a source of this uncertainty would be the drag in an experiment designed to measure the acceleration of gravity near the earth’s surface. The commonly used gravitational acceleration of 9.8 m/s2 ignores the effects of air resistance, but the air resistance for the object could be measured and incorporated into the experiment to reduce the resulting uncertainty in the calculation of the gravitational acceleration.

I’ll have a little more to say about this distinction between aleatoric and epistemic uncertainty—which I presume implies corresponding aleatoric and epistemic probabilities—a few paragraphs down. For now, notice that the context seems a practical one, and in fact seems oriented towards risk management. Also notice that, technically, aleatoric uncertainty seems something of a special case of epistemic uncertainty.

Return now to our starting place, the entry Probability Interpretations. The next section I’d like to look at is titled “Logical, Epistemic, and Inductive Probability,” which features the following passage:

It is widely recognized that the term “probability” is sometimes used in contexts where it has nothing to do with physical randomness. Consider, for example, the claim that the extinction of the dinosaurs was probably caused by a large meteorite hitting the earth. Statements such as “Hypothesis H is probably true” have been interpreted to mean that the (presently available) empirical evidence (E, say) supports H to a high degree. This degree of support of H by E has been called the logical probability of H given E, or the epistemic probability of H given E, or the inductive probability of H given E.

The differences between these interpretations are rather small, and may seem inconsequential. One of the main points of disagreement lies in the relation between probability and belief. …

The entry goes on to very briefly note potential points of disagreement about subtle distinctions between those concepts, which I won’t worry about. I’ll simply assume this to be a standard Bayesian take on epistemic probability, and will note that it again points to Ramsey as a key thinker in the development of the idea, noting that “Ramsey held that epistemic probabilities simply are degrees of rational belief, rather than being logical relations that merely constrain degrees of rational belief.”

This notion of “degrees of belief” can, I think, be fairly summarized by way of example. When you flip a fair coin, your degree of belief that it lands Heads is 0.5. In other words, it is determined by the probability of the coin landing Heads. (This basic idea also underlies Bayesian epistemology.7) In most cases in life, however, probabilities are not so clear cut. So sometimes your degree of belief—which is to say, the probability you assign an event—may be to some extent subjective. This gets at the most controversial aspect of Bayesian probability, which I won’t get into here. But I will note that, these days, most people are on board with Bayesian probability to some significant extent (or even to a point of ecstatic dogma), though there does remain the question of whether, in some important sense, all probability, as practiced by imperfect mortals, is epistemic.

I’ll assume that it’s possible for probability to be epistemic even in a classical (i.e., non-Bayesian) picture, though I’m not sure what all this means for randomness. I’ll table that question for now.

The third relevant section of the entry Probability Interpretations is titled “Frequentism,” and says the following about aleatory probability:

Frequentists posit that the probability of an event is its relative frequency over time, i.e., its relative frequency of occurrence after repeating a process a large number of times under similar conditions. This is also known as aleatory probability. The events are assumed to be governed by some random physical phenomena, which are either phenomena that are predictable, in principle, with sufficient information (see determinism); or phenomena which are essentially unpredictable. Examples of the first kind include tossing dice or spinning a roulette wheel; an example of the second kind is radioactive decay. In the case of tossing a fair coin, frequentists say that the probability of getting a heads is 1/2, not because there are two equally likely outcomes but because repeated series of large numbers of trials demonstrate that the empirical frequency converges to the limit 1/2 as the number of trials goes to infinity.

Compare this with the earlier-noted definition of “aleatory” uncertainty (that was called “aleatoric,” but the words should mean the same thing; see the below note on the German translation of the word). Interestingly, this section links to a full article on Frequentist Probability that makes no mention of the words “aleatory” or “aleatoric” (though does refer to Bayesian/subjective probability).

The terms “ontic” and “ontological” do not show up in any of the above-referenced Wikipedia entries. Nor do they appear in the Wikipedia entry on Randomness (more about which below), though the “See Also” section (like most “See Also” sections here) has a link called “Aleatory” that leads to an entry titled Aleatoricism, a term defined as: “the incorporation of chance into the process of creation, especially the creation of art or media. The word derives from the Latin word alea, the rolling of dice.” I’m most familiar with such practices in the context of aleatoric or “chance” music, such as that associated with John Cage (this topic too, of course, has its own Wikipedia entry: Aleatoric Music, where it is noted that the introduction of the word into English came from a confused translator who transliterated the the German terms “Aleatorik” and “aleatorisch” into English as “aleatoric,” rather than use the existing adjective “aleatory.” “More recently,” it’s noted, “the variant ‘aleatoriality’ has been introduced.8).

Ok, I think we’re getting the general idea. Some of these terms are used loosely or in overlapping, though not identical, ways, and this blurring of usage corresponds (at times) to a kind of conceptual blurring of the phenomena denoted by those terms. Of course, we’ve come nowhere near exhausting the appearance of these terms at Wikipedia. Nor will we today. Though I will look briefly at three more entries before moving to the next source.

The entry Random Variable notes that “random variables” may be referred to as “aleatory variables,” though I take it this is pretty rare (besides, random variables aren’t actually random, and they’re not really variables—rather, they’re functions that map each possible outcome of a sample space to a number).

There are two entries to look at that deal in randomness. The first is, unsurprisingly, the one on Randomness. It doesn’t mention epistemic randomness, but does refer—in the section “In Science,” subsection “In the Physical Sciences”—to objective randomness, which I presume to be synonymous with my usage of “ontic randomness”:

According to several standard interpretations of quantum mechanics, microscopic phenomena are objectively random. That is, in an experiment that controls all causally relevant parameters, some aspects of the outcome still vary randomly. For example, if a single unstable atom is placed in a controlled environment, it cannot be predicted how long it will take for the atom to decay—only the probability of decay in a given time. Thus, quantum mechanics does not specify the outcome of individual experiments but only the probabilities. Hidden variable theories9 reject the view that nature contains irreducible randomness: such theories posit that in the processes that appear random, properties with a certain statistical distribution are at work behind the scenes, determining the outcome in each case.

Fair enough, but I’d be careful with phrases like “does not specify the outcome,” because the general idea in quantum mechanics is that the prediction can be extremely accurate, maybe perfectly accurate, even though the prediction is probabilistic. Terms like “accurate prediction” and “objectively probabilistic,” or, if you prefer, “irreducibly random,” are not at odds with one another.

The other entry—Interpretations of Quantum Mechanics—mentions the epistemic-ontic distinction directly. It doesn’t do this explicitly in the context of probability or randomness, but certain has implications for those things. It is noted at the beginning of the entry, for example, that competing interpretations of quantum mechanics

differ on such fundamental questions as whether quantum mechanics is deterministic or random, which elements of quantum mechanics can be considered “real,” and what is the nature of measurement, among other matters.

I will resist the temptation to get into the topics to which this may lead us, regarding, for example, the so-called measurement problem involving neat stuff like wave function collapse and other questions about the relation between the models of quantum mechanics and the actual world. Whether or not wave function collapse is inherently probabilistic is not at issue. What is at issue is how we may talk about that sort of phenomenon, whether or not it exists, in the context of probability and randomness. I call this “ontic” probability/randomness.10

A little further down, in the section “Nature,” it is noted that two qualities vary among interpretations of quantum mechanics:

(1) Ontology—claims about what things, such as categories and entities, exist in the world

(2) Epistemology—claims about the possibility, scope, and means toward relevant knowledge of the world

In philosophy of science, the distinction of knowledge versus reality is termed epistemic versus ontic. A general law is a regularity of outcomes (epistemic), whereas a causal mechanism may regulate the outcomes (ontic). A phenomenon can receive interpretation either ontic or epistemic. For instance, indeterminism may be attributed to limitations of human observation and perception (epistemic), or may be explained as a real existing maybe encoded in the universe (ontic). Confusing the epistemic with the ontic, like if one were to presume that a general law actually “governs” outcomes—and that the statement of a regularity has the role of a causal mechanism—is a category mistake.

In a broad sense, scientific theory can be viewed as offering scientific realism—approximately true description or explanation of the natural world—or might be perceived with antirealism. A realist stance seeks the epistemic and the ontic, whereas an antirealist stance seeks epistemic but not the ontic. In the 20th century’s first half, antirealism was mainly logical positivism, which sought to exclude unobservable aspects of reality from scientific theory.

This, again, starts to get into areas that are not directly related to probability and raises yet more questions (for example, are claims of scientific realism about science as it is right now or about where the overall Human Project of Science, as we currently understand that project, will take us if we don’t blow ourselves up first?). I’ll step past such questions, but I think they’re worth letting seep over the present lines of discourse to remind us that these conversations have been going on as part of a larger, and very complicated, discussion that is far from finished.

That’ll do for the beautiful rabbit hole we call Wikipedia. Well, for now. I’m sure we’ll find ourselves back there soon.

3.3 Stanford Encyclopedia of Philosophy (SEP)

SEP’s articles are written not as a community Wiki, but by experts in the relevant subfield who sign their names as authors. The principle article I’ll look at is the entry, Interpretations of Probability by Alan Hájek (a name commonly encountered in the word of philosophy of probability).

“Subjective probability” is mentioned several times and is introduced thus:

We may characterize subjectivism (also known as personalism and subjective Bayesianism) with the slogan: “Probability is degree of belief.” … Beginning with Ramsey 11, various subjectivists have wanted to assimilate probability to logic by portraying probability as the logic of partial belief. (Section 3.3).

“Epistemic probability” is mentioned once, in Section 4, and seems to be more or less taken granted to meaning what one expects: the probability one is best off assigning given incomplete knowledge about the “real” or “objective” probability. Phrases like “the way in which such ignorance should be epistemically managed” (Section 3.1) show up elsewhere.

“Objective probability” is brought up several times, including in the formulations “non-fundamental objective probabilities” (Section 4) and “objective chance function” (Section 3.3.4) and simply “objective chance” (Sections 3.3.4 and 4). At any rate, stated without qualification, “objective probability” seems to be meant just as it sounds (I won’t comment on the other formulations, especially “objective chance function,” which has a technical definition due to David Lewis).

Another SEP entry worth looking at is Rudolf Schuessler’s Probability in Medieval and Renaissance Philosophy, which has a section titled “Subjectivity and Objectivity” (Section 4.4). Here’s a passage:

The question of whether medieval forms of probability were subjective or objective has long puzzled modern researchers. It is difficult to come up with an answer, not least because medieval authors did not employ the (characteristically modern) terminology of subjectivity and objectivity with respect to judgments of probability. Nevertheless, the discussion above of medieval uses of probability-related terms sheds some light on matters of subjectivity and objectivity.

As has been shown, probabilis or verisimilis were predicates for the qualitative support a body of evidence gave to the truth of a proposition or the fittingness of a sign. We have already seen what kinds of evidence for probability were accepted at the time. A subjective aspect of probability ascriptions thus consisted in a belief obtained by the supporting relationship in a particular case. An objective aspect depended on the actual existence of the supporting basis.

This kind of objectivity does not make probability a feature of the world or a theoretical construct based on facts of nature. …

I take it that, here, “this kind of objectivity” is not ontic (i.e., “does not make probability a feature of the world”). In my own usage, I’d like to use different words for these concepts, where the “real” probability of something is a matter of building the best possible probability model according to known facts. For example, when the coin has a probability of 0.5012 of landing Heads according to a frequency measurement over thousands of flips. That becomes a general model for how the coin will behave when, say, vigorously flipped by a human hand in a typical environment (and not, say, by a robot designed to flip it the exact same way every time, landing it on the exact same surface every time, etc.). In other words, even if you could exactly predict what a coin will do this time, given certain conditions, this doesn’t mean you know what it will do given slightly different conditions. So, we build a general model of 50–50 for all two-sided coins, but could build a specific general model for any given coin, according to some standard of measurement. According that measurement with a frequentist picture may be problematic because the coin could be physically altered after thousands of flips (the edges might get smoother). So we might want to develop a different system—say, by some sophisticated analysis of the coin’s physical dimensions, weight distribution. My point is that saying what a “real” probability is not such an easy thing (even once we’ve agreed on a model; see Hanna’s article, below), but it make sense theoretically and has some approximation in practice that distinguishes it from my meaning of “ontic” probability.

In another article—Quantum Approaches to Consciousness by Harald Atmanspacher—such clarification is achieved by referring to what I call ontic probability as quantum probability. In the Introduction:

Quantum theory introduced an element of randomness standing out against the previous deterministic worldview, in which randomness, if it occurred at all, simply indicated our ignorance of a more detailed description (as in statistical physics). In sharp contrast to such epistemic randomness, quantum randomness in processes such as spontaneous emission of light, radioactive decay, or other examples of state reduction was considered a fundamental feature of nature, independent of our ignorance or knowledge. To be precise, this feature refers to individual quantum events, whereas the behavior of ensembles of such events is statistically determined. The indeterminism of individual quantum events is constrained by statistical laws.

As far as I know, the notion of ontologically embedded probability originates with quantum theory. Famously, Einstein rejected this aspect of the theory, which I understand to fall under what we now refer to as the Copenhagen interpretation of quantum mechanics. See again my above mention of wave function collapse for more on this, as well as footnote #10 pointing to Sean Carroll’s lecture series, in which Carroll makes a good argument for rejecting this interpretation as the best model of reality. Here, again, I’m not interested in trying to parse out the details distinguishing competing interpretations (the other being a family of “many-worlds” theories), but rather would like to understand how folks are using their terms. That said, I’m not so sure “quantum” probability suits my purposes as quantum theory doesn’t just describe the behavior of sub-atomic world, but describes also the world overall, as it is the composition of the sub-atomic world. This, too, is something Carroll points out, and is apparently part of why—or is least bound up with why—he (and, it seems, the subset of physicists who spend the most time thinking about this question) rejects the Copenhagen interpretation. Plus, if the Copenhagen interpretation is wrong, the word “quantum” will be misleading.

Ok, there are, of course, lots more articles at SEP that use these terms. But I think we get the idea.

Now, how about usage among philosophically minded folks writing about probability? I’ll look at five core sources (along with whatever additional sources those lead me to). The first four will be (recommendable) books I grab from my collection; the fifth will be an academic article.

3.4 The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference by Ian Hacking (first published in 1975, then with a new Introduction in 2006).

Chakraborty remarks in his paper that Hacking is often credited for coining the term “epistemic epistemology,” and refers us to this book. It’s a classic of the field, and rightly so. Though the term, as Hacking uses it, certainly existed before 1975. Though I wouldn’t presume that all uses, past or present, are the same as that employed by Hacking.

In the Wikipedia entry dedicated to Frank P. Ramsey, for example, it’s noted that Ramsey disagreed with John Maynard Keynes, who, in his 1921 A Treatise on Probability , “argued against the subjective approach in epistemic probabilities.” In that book, Keynes uses the term “subjective,” though only once followed by the word “probability” (that I could find):

Cournot, in his work on Probability, which after a long period of neglect has come into high favour with a modern school of thought in France, distinguishes between “subjective probability” based on ignorance and “objective probability” based on the calculation of “objective possibilities,” an “objective possibility” being a chance event brought about by the combination or convergence of phenomena belonging to independent series. (p. 323)

I’m not totally sure what it meant here by “objective probability,” though it sounds close to what I’ve been calling the “real” probability than to one that’s “ontic.”

Keynes doesn’t mention “epistemic probability” explicitly, but does write things like:

The question of epistemology, which is before us, is this: Is it necessary, in order that we may have an initial probability, that the object of our hypothesis should resemble in every relevant particular some one object which we know to exist, or is it sufficient that we should know instances of all its supposed qualities, though never in combination? (p 344)

The earliest example of explicit use that I could find of “epistemic probability” is in the 1943 article, “Existential and Epistemic Probability” by philosopher David Hawkins 12 Is Hawkins’ use of these terms self-explanatory? For “existential probability,” he gives a definition that I think, again, aligns with other the notion of “real” probability, but not “ontic”:

Existential probability refers to the probability that events occur or that propositions about the occurrence of events are true. Propositions about existential probabilities are propositions about physical realities. They have to do with the metrical properties and causal relations of chance mechanisms, and with the relative frequencies of different kinds of events produced by those mechanisms. (p. 225)

For “epistemic probability,” he provides a familiar definition:

Epistemic probability refers to the probability that beliefs or inferences are correct. Propositions about epistemic probabilities are propositions about the weight of evidence and the reliability of our beliefs. They have to do with the procedures of inference. (p. 225)

Hawkins then provides an interesting note about the relationship between these ideas:

Inferences about existential probabilities are themselves matters of epistemic probability, since the latter are statements about physical realities not altogether subject to direct perceptual verification. On the other hand at least some epistemic probabilities are derivative from those which are existential, since prediction based upon the latter is itself a mode of probable inference.

This is precisely a point I myself intend to make below, when attempting to isolate a key point about the distinction between ontic and epistemic probability on the one hand and ontic and epistemic randomness on the other. I’ll call back to this later. The point now is that the notion of “epistemic” probability predates Hacking’s book, which doesn’t reference Hawkins (who remarks in his third footnote that there are other possible interpretations of the term “epistemic probability”).

According to Google’s Ngram Viewer, Hawkins’ paper is cited on occasion between 1943 and the publication of Hacking’s book in 1975. The term “epistemic probability” also comes up in at least two other sources. One is from page 263 in a 1959 issue (volume 40) of the journal Anthropological Linguistics, in which an example of “epistemic probability” is offered: “the trip should take about fourteen days.”

The other is from what I assume to be the first edition of Brian Skyrms book, Choice and Chance: An Introduction to Inductive Logic (now in it’s fourth edition). I haven’t read it, but I see that there’s a section of Chapter 2 called “Epistemic Probability,” were that term is contrasted with the “inductive probability of an argument,” which is “a measure of the strength of the evidence that the premises provided for the conclusion”; in contrast, it’s not correct to think of the premises or conclusion themselves—i.e., statements—in terms of inductive probability:

There is, however, some sense of probability in which it is intuitively acceptable to speak of the probability of a premise or conclusion. When we said that it is improbable that there is a 2000-year-old man in Cleveland, we were relying on some such intuitive sense of probability. There must then be a type of probability, other than inductive probability, that applies to statements rather than arguments.

Let us call this type of probability “epistemic probability” because the Greek stem “episteme” means knowledge, and the epistemic probability statement depends on just what our stock of relevant knowledge is. Thus, the epistemic probability of a statement can vary from person to person and from time to time, since different people have different stocks of knowledge at different times. (Skyrms 23)

This is a reasonable and, by now, familiar definition of “epistemic probability.” Skyrms point about the subjective nature of probability assignments is well taken: if I conceal a coin from your view and I know it’s sitting Tails-up, the probability for me that it’s sitting Heads-up is zero, but for you it’s 1/2. Also interesting is the mention of intuition.

Now, finally, to Hacking. The goal here is to get a sense of how he meant the terms in question. I’ll reproduce relevant passages, most of which require no additional commentary, in their order of appearance. Note that the word “ontic” does not appear in the book:

… the probability emerging at the time of Pascal is essentially dual. It has to do with stable frequencies and with degrees of belief. It is, as I shall put it, both aleatory and epistemological. (Hacking 10) …

It is notable that the probability that emerged so suddenly [around 1660] is Janus-faced. On the one side it is statistical, concerning itself with stochastic laws of chance processes. On the other side it is epistemological, dedicated to assessing reasonable degrees of belief in propositions quite devoid of statistical background. The duality of probability will be confirmed by our detailed study of the history between 1650 and 1700. Even now it is clear enough. Pascal himself is representative. His famous correspondence with Fermat discusses the division problem, a question about dividing stakes in a game of chance that has been interrupted. The problem is entirely aleatory in nature. His decision-theoretic argument for belief in the existence of God is not. It is no matter of chance whether or not God exists, but it is still a question of reasonable belief and action to which a new probable reasoning can be applied. (Hacking 12) …

Hence, we must not ask, “How on the one hand did epistemic probability become possible and how, on the other, did calculations on random chances become possible?” We must ask how this dual concept of probability became possible. (Hacking 12)

These passages set the stage for how Hacking will go on to use the words “aleatory” and “epistemic.” He points out that the above duality has been “long known to philosophers” (Hacking 13), and gives examples of other terms proposed, such as Condorcet’s suggestion of “facilité for the aleatory concept and motif de croire for the epistemic one” (Hacking 13). He returns to this duality repeatedly in the book, and of course delves with greater subtlety into the topic than I have space (or need) to get into here. It seems to me that “aleatory” essentially means “real” probability, but will look at a few more interesting experts to bear this out:

Chances, odds, “hazards” (the stock in trade of aleatory probability) are basically quantitative. There is no way to understand odds without understanding numerical ratios. Epistemic probability is not like this. (Hacking 73)

This seems to get at the important point that epistemic probabilities need not be numerical, something that is a central idea to another book I’ll look at here, Franklin’s The Science of Conjecture. Indeed, Hacking goes on to note that epistemic probability can involve—sometimes necessitates—qualitative, rather than number-based, comparisons of probabilities and “the degree to which evidence warrants several propositions” (Hacking 73; Hacking finds support here, by the way, from the aforementioned Keynes treatise). “But,” he continues, “as a matter of historical fact epistemic probability did not emerge as a significant concept for logic until people thought of measuring it” (Hacking 73). To investigate that emergence, he begins with when the word “probability” seems to have been first used to refer to something measurable: “The answer seems to be 1662, in the concluding pages of the Port Royal Logic” (Hacking 73). That’s further back than I’m going to venture here, but it gives a nice sense of the depth of Hacking’s inquiry.

Perhaps his broadest, and most succinct, explanation of the duality is the following:

Aleatory probabilities have to do with the physical states of coins or mortal humans. Epistemic probabilities concern our knowledge. (Hacking 123)

This is certainly simpler, though also less thought-provoking, than the one he gives on the next page, after a brief discussion of the de re/de dicto distinction:13

But at a very gross level we may, with Buoudot14 say that aleatory probability is de re, having to do with the physical characteristics of things, while epistemic probability is de dicto, for it concerns what we know and hence what can be expressed by propositions. (Hacking 124)

I feel we have by now skimmed the surface just enough to see that, while Hacking explores these concepts from multiple perspectives thus leaving room for much more reflection and discussion (including whether we should even be using the word “probability” in both aleatory and epistemic contexts), his core understanding of the terms “aleatory” and “epistemic” align well enough with the two concepts I noted above.

He also seems to use, or at least recognize that many people tend to use, the terms “epistemic,” “subjective,” “personalist,” “Bayesian,” and “inductive” probability roughly synonymously, as when he writes of “what we now call the inductive or epistemological or subjective probability…” (Hacking 133). A deeper look into Hacking’s book would reveal further subtleties, and, for example, observations about finer distinctions of usages depending on the era and thinkers he’s discussing in a given passage (see for example the paragraphs I share below on the word “subjective”) or chapter (e.g., in those titled “Inductive Logic” and “Induction”). There are other related terms that come up in the book, such as of “logical” and “rational” probabilities, but it doesn’t seem helpful at this point to discuss all of them. I’m content that we’ve sufficiently covered the core concepts that compose the aleatory-epistemic duality with which he is concerned. Though perhaps a real-world example would help round this out:

Look no further than the vexed debate between clinical medicine and so-called evidence-based medicine. (There’s a name chosen for rhetorical purposes if ever there was one!) Evidence-based medicine means frequencies and randomised trials; clinical medicine means the formation of coherent degrees of belief. The same old duality. Evidence-based medicine will win out, but not because of good (inductive) reason. (Hacking, from the next-to-last page of the unnumbered 2006 Introduction)

To be clear, Hacking notes that most practitioners go about their business of doing and talking about statistics without paying attention to the duality, but that

…there are a few extremists on either side. There are the personalists, including de Finetti, who have said that propensities and statistical frequencies are some sort of ‘mysterious pseudo-property’, that can be made sense of only through personal probability. There are frequentists who contend that frequency concepts are the only ones that are viable. (Hacking 14)

I bet there are even fewer hardcore frequentists today than at the time Hacking wrote the book, and also more Bayesian extremists, which is in essence the same thing as subjectivism or personalism. At least, that is, among practitioners, as it seems philosophers of probability and statistics have for a good while leaned Bayesian, though may be growing, as a group, more sympathetic to the frequentist view (see, for example, this post from philosopher of statistics Deborah Mayo: “Frequentists in Exile,” who, by the way, has recently written a book I’ve just started reading, Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars. More on this another day.)

I also bet there is much more awareness of the distinction among practitioners, and that this is in large part a consequence of the so-called replication crisis that’s been getting increasing attention over the last several years, especially, it seems to me, since about 2011 (I associate the broader phenomenon taking the shape we now call the “replication crisis” with the 2011 publication of Daryl Bem’s article claiming to have produced statistically significant results in favor of ESP; this at least seems to be where widespread worries about methodology start to take the form of what would soon fall under the umbrella label “replication crisis,” though it is not mentioned at the topic’s Wikipedia entry: Replication Crisis. For an quick and thought-provoking account of the phenomenon, see this two-part series from the Hi-Phi Nation podcast (a show I value for its regular inclusion of philosophers among its guest experts): Hackedmics I: The Control [Episode 6, 3/6/17; Centers on ESP as studied by scientists, culminating with Bem’s article]; Hackademics II: The Hackers [Episode 7, 3/14/17; Centers on Brian Nosek et al’s efforts that resulted in the concretization of the phenomenon as we think of it today]).

Try listening to a podcast featuring social psychologists, and you’ll commonly hear Bayesian language, sometimes explicitly contrasted with the concepts of “classical” or “frequentist” statistics. As far as I can tell most folks these days are on board with Bayesian methods even if they haven’t transition (I commonly hear older professors say stuff like, “My grad students are teaching me Bayesian statistics”)—though some more are more deeply Bayesian than others, to a point of being part of their identity. I recently heard it commented by a wife of a data scientist that he worked the name “Bayes” into his wedding vows.

I’m now reminded of the tongue-in-cheek, vaguely (self-)mocking introductory notes in Eliezer Yudkowsky’s explanation of Bayes’ theorem, a simple equation that provides the core of Bayesian statistics and that, in isolation, just “looks like this random statistics thing,” as Yudkowsky puts it. He continues:

So you came here. Maybe you don’t understand what the equation says. Maybe you understand it in theory, but every time you try to apply it in practice you get mixed up trying to remember the difference between p(a|x) and p(x|a), and whether p(a)p(x|a) belongs in the numerator or the denominator. Maybe you see the theorem, and you understand the theorem, and you can use the theorem, but you can’t understand why your friends and/or research colleagues seem to think it’s the secret of the universe. Maybe your friends are all wearing Bayes’ Theorem T-shirts, and you’re feeling left out. Maybe you’re a girl looking for a boyfriend, but the boy you’re interested in refuses to date anyone who “isn’t Bayesian.” What matters is that Bayes is cool, and if you don’t know Bayes, you aren’t cool.

Why does a mathematical concept generate this strange enthusiasm in its students? What is the so-called Bayesian Revolution now sweeping through the sciences, which claims to subsume even the experimental method itself as a special case? What is the secret that the adherents of Bayes know? What is the light that they have seen?

Soon you will know. Soon you will be one of us. (From Yudkowsky’s explanation of Bayes’ theorem.)

Finally, Hacking does not employ the word “ontic,” but he does reference the notion of what we can think of as truly or fundamentally or irreducibly probabilistic or random—what above was referred to as “quantum” probability. Hacking makes reference to such phenomena in the usual way; that is, in the context of physics:

We also live in a world that is, from the point of view of fundamental physics, necessarily and essentially a matter of probabilities. (Hacking, fourth page of 2006 Introduction) …

Suppose that some evidence counts for r and some counts against it. Nowadays most of us accept that the relation between some propositions and others may be fundamentally probabilistic. We acknowledge this because we have to accept as an irreducible fact, that a quantum system in state Ψ confers a 60% probability on the proposition that a particle is in region r. According to the formalism of quantum mechanics Ψ cannot be decomposed into’ Ψ1 or Ψ2 with Ψ1 entailing that the particle is in r and Ψ2 entailing the opposite. Of course very similar facts have been familiar in everyday life, although only microphysics makes some people notice them. The fact, ϕ, that the rear tires are completely bald, confers 60% probability on having a flat before reaching Massachusetts. It is a mere myth that ϕ should break down into a set of cases ϕ1, which entail getting a flat tire, and another set ϕ2, entailing the opposite. We certainly know of no such decomposition and have no good reason to think there is one. Yet the myth that every problem in probability can be reduced to a set of favourable and unfavourable cases persisted for centuries. One reason for this is obvious: the analysis fits many games of chance. (Hacking 90–91)

Interestingly, the second passage is from a chapter called “Probability and the Law.” If I understand it, he means to say that, when the randomness of an event—let’s call it E—is fundamentally probabilistic (he only uses this term once in the book; it’s what I call “ontic”), given an identical prior state, ᴪ, of multiple experiments, 60% of the time E will occur and 40% of the time E will not occur. This would be like the following. You flip a coin and it lands Tails. We rewind the tape of the universe to a nanosecond before you flipped the coin and replay, but this time it lands Heads. Suppose that half the time we do this, it lands Tails. As I’ll discuss in further depth below, real coin flips aren’t like this: they are deterministic so that you’d get the same result every time given identical prior conditions. It also means you could predict the flip result given enough information about those conditions, and the computing capacity to process that information, etc.15

But with the imagined ontic coin, we can’t stipulate initial conditions State-1, which renders Tails, and State-0, which renders Heads, because the outcome is fundamentally probabilistic, or we might say “indeterministic.” It’s just essential to the event that it’s going to come out Tails half the time, even given identical starting conditions.

We might want to tell a more nuanced story of what that 50% means (e.g., degree of credence in the coin’s landing Tails), but I think this is a fine way to wrap one’s mind around the idea.

Hacking seems (again, if I understand him) to imply the same in his speculation about our tendency to generalize from chance games, but seems to be putting the far more complex event of driving with bald tires in the same category as ontic randomness. It’s not obvious to me that this is correct, and that what’s really going on is that the event’s complexity simply renders us ignorant. While I think the difficulty of prediction there is due to epistemic limitations in the face of complexity alone, Hacking seems to agree with this but with the proviso that, were an omnipotent consultant to weigh in on whether the tires will blow, she’d assign a probability between 0 and 1, contrary to my “merely too complex” view.

In other words, we’d assign an epistemic probability while the true probability follows an ontic standard. We might get our epistemic probability from a whole lot of data, e.g., we’ve collected years worth of such incidents and processed it according to things like tire brand and age, road conditions, driver’s age, and as many relevant variables as you like. This would tell a frequentist story: how often does that happen given these conditions. But if this produces the true probability, it’s due to luck. Again, if it’s a deterministic world, the true probability is 0 or 1; if it’s indeterministic (i.e., “ontic”) it could be between 0 and 1. The concern this raises for me is how to distinguish my thinking about epistemic probability and statistically based probabilities—which I think will in general be referred to by authors here as “aleatory,” and which I referred to before as needing to conform to some (perhaps arbitrary) model standard, e.g. frequentist. Well, as has already been pointed out: perhaps all probability as practiced by humans comes down to epistemic probability.

This in mind, it is interesting to examine Hacking’s discussion of three applications of the term “subjective” in probability theory. I’m skipping over a lot of fascinating discussion here, but this gets the gist across (note his comment that it’s worth getting the terminology straight!):

Where are probability and induction today? Two different things seem to have evolved. One is the theory of consistent beliefs which F. P. Ramsey was the first to envisage in its depth, in 1930. That is now called ‘Bayesian’, although in my opinion it has little to do with Thomas Bayes. The other is the application of the theory of stable relative frequencies to real-world prediction. (Hacking penultimate and next-to-last pages of 2006 Introduction) …

…the most recent fashion, pioneered by F. P. Ramsey in 1926, and winning wide recognition after the book by L. J. Savage published in 1954. Some have called this subjectivism; Savage called it personalism. Most statisticians call it the “Bayesian” theory. (Hacking 15) …

Bernoulli’s originality is to see what the notion of certainty implies for probability. He thinks certainty is of two sorts, Subjective and objective. Anything that will occur is already objectively certain. An historical dictionary gives a good reminder of what is going on here: the word ‘certain’ once meant what was decided by the gods. If some event were objectively uncertain then the gods could not have made up their minds. Perhaps in a more mechanistic age, if what happens does not happen with certainty then determinism is false. (Hacking 145–146) …

Bernoulli undoubtedly imported the word “subjective” into probability theory but by now the word in this connection has become equivocal. Several distinct modern theories of probability have been called subjective, and since there has been so much idle controversy about them it is worth getting the terminology straight. For my part I prefer to avoid the word ‘subjective’ altogether, so as to avoid potential ambiguity, but since Bernoulli started the trend we cannot evade it here. In recent writing three different kinds of probability have been called subjective. The most extreme subjectivism is that of Bruno de Finetti. L. J. Savage has felicitously called it ‘personalism.’ Then there are the theories of logical or inductive probability developed by J. M. Keynes and others, and which, though insisting on a measure of objectivity, are often called ‘subjective’ by their detractors. Thirdly there is yet another concept of subjectivity current among many present philosophers of quantum physics.

A convenient indicator is available to test an allegedly subjective theory. It devolves on the notion of unknown probabilities. In the extreme personalist theory, probabilities may be unknown only insofar as one ‘fails to know one’s own mind’ (to use Savage’s phrase). In the logical theory probabilities may be unknown by failure to do logic but no experiment will help check up on logical probability. Finally in some physicists’ use of the term ‘subjective’, subjective probabilities can actually be checked by experiment. We shall find that, contrary to the authorities mentioned in the beginning of this chapter, Bernoulli’s subjectivism is less like the personalist or logical point of view, and more like that of the physicists. (Hacking 146–147) …

… as Heisenberg puts it, ‘the probability function contains the objective element of tendency and the subjective element of incomplete knowledge’.16 Heisenberg writes not of my fiction of coins and urns, but of an essential feature of the quantum theory. As he interprets the theory, there exist systems in pure states whose probability function is ‘completely objective’. But there also exist ensembles of systems in different pure states; these are called mixtures. Probability statements about a system in a mixture depend on ‘statements about our knowledge of the system, which of course are subjective’. Subjective, yes, but no mere matter of opinion, for these so-called subjective probabilities can still, he says, ‘be checked by repeating the experiment many times’.

De Finetti means something rather different. He views subjective probability as an indicator of purely personal degrees of belief and he has discovered useful constraints on them. For all his constraints I may in consistency have virtually any personal probability function for any given set of alternatives. Probabilities are unknown only insofar as I may fail to know my own mind.

Intermediate is the theory of logical or inductive probability according to which, any body of evidence e uniquely determines a probability for any hypothesis h. This is best represented as a function… (Hacking 148)

The above comments from Heisenberg echo my earlier noted concerns about the frequentist picture, involving running an experiment multiple times, coming down to an epistemic probability, that may or may not reveal the true probability, which may or may not be ontic. Uncovering this true probability, to the greatest possible degree of precision, may be out of reach. For example, we might say you have to run the experiment infinitely many times, which is physically impossible to do. Even running it millions of times might be impossible due, for example, to the coin’s edge wearing down. We could perhaps reproduce the coin repeatedly down to the finest possible physical detail, but this won’t necessarily do it either, as the randomness of the coin flip includes its environment. Again, if you flip a coin the exact same way in the exact same environment every time, it will land the same way every time (this has been demonstrated by Persi Diaconis et al, which I’ll say more about below).

There may be some experiments that really can be thus repeated, of the sort interesting to particle physics, such that even the exact same environment leads to different results (due to being an ontic process). But the question of how many experiments enough persists. I don’t know how physicists answer this question, and with what degree of consensus; I’ll save investigation of this for another day.

I’ve encountered the idea of ontic probability in fields other than quantum mechanics (the flat tire example aside), especially biology and especially as biology pertains to operations of the mind and free will. It most commonly comes up in my own reading in the context of quantum mechanics, however.

As with all the sources here, I could spend the next year revisiting Hacking’s book and considering terminology (e.g., “pure” and “mixed” also seem to be possible replacements of “ontic” and “epistemic,” respectively, starting with “Leibniz’s notion of pure and mixed proofs in law” [Hacking 151]). But I’ll move on now, leaving this book with one last passage, which again notes that some physicists push against the idea of ontic randomness in quantum mechanics:

Only the advent of the quantum theory has made it possible to conceive of statistical regularity as a brute and irreducible fact of nature. And there is still a small but valiant programme of ‘hidden variable’ theorists who contend that the gross statistical laws must be the manifestion of some as yet unspecified deterministic laws.

Thus the belief in an omnipresent deity that maintains mean statistical values has a strong and lasting effect on the aleatory side of probability. The central doctrine is that statistical laws merely describe constant regularities. Just like gravity, they do not get at efficient causes. This conception of ‘mere regularity’ is important not only for the aleatory side of probability, but also for the epistemic side. It is a final ingredient for the sceptical problem of induction, stated by Hume in 1739. (Hacking 174–175)

3.4 The Science of Conjecture: Evidence and Probability before Pascal by James Franklin (2001; new preface added in 2015)

Franklin’s research interests include philosophy of mathematics and the history of ideas.17 He provides some useful definitions in the 2001 preface of The Science of Conjecture (an incredibly well-researched and philosophically rich book; an invaluable resource for anyone interested in the history of probabilistic reasoning):

Probability is of two kinds. There is factual or stochastic or aleatory probability, dealing with chance setups such as dice throwing and coin tossing, which produce characteristic random or patternless sequences. Almost always in a long sequence of coin tosses there are about half heads and half tails, but the order of heads and tails does not follow any pattern. On the other hand, there is logical or epistemic probability, or non-deductive logic, concerned with the relation of partial support or confirmation, short of strict entailment, between one proposition and another. A concept of logical probability is employed when one says that, on present evidence, the steady-state theory of the universe is less probable than the big bang theory or that an accused’s guilt is proved beyond reasonable doubt though it is not absolutely certain. How probable a hypothesis is, on given evidence, determines the degree of belief it is rational to have in that hypothesis, if that is all the evidence one has that is relevant to it.

It is a matter of heated philosophical dispute whether one of these notions is reducible to the other. (Franklin, 2001 Preface, pages unnumbered)

These words and definitions are familiar by now. In particularly, I find his mention of “aleatory” probability useful. Note also his listing here of “epistemic” and “logical” probability as synonymous; he will complicate these and other above-mentioned synonyms later (namely, see the four factors of confusion listed below). Franklin gives two references for his claim about there being two kinds of probability: Hacking’s Emergence of Probability (see above) and David Stove’s 1973 book, Probability and Hume’s Inductive Scepticism (I haven’t read it).

And here are a few illuminating excerpts from the 2015 Preface. Note, for example, his emphasis on the importance of non-numerical probabilities in the logical (or subjective) case, which I alluded to earlier as a central theme of Franklin’s book. I am excluding several citations here (of which this book is filled to the brim):

[The Science of Conjecture]’s conception of the subject matter stemmed from an objective Bayesian (or logical probabilist) theory of probability. According to that theory, as developed by Keynes in his Treatise on Probability and by later authors 18 the main notion of probability is of an objective logical relation holding between a body of evidence and a conclusion. The body of evidence available in court does or does not make the defendant’s guilt highly probable; the known facts do or do not support the theory of global warming, irrespective of any contingent facts about the world or what anyone’s opinion is. Logical probability may or may not be numerical; even if it is, qualitative or approximate judgments are often of most importance.

That perspective opened up all kinds of evaluation of uncertain evidence as the natural subject matter of a history of probability. Thus, The Science of Conjecture focused on the law of evidence, which, over many centuries of thought, especially in medieval Roman law, had developed evidential concepts like the modern proof beyond reasonable doubt. Moral theory and business were also familiar with concepts of probabilities and risks, mostly quantified only loosely.

During the late twentieth century, debate on “interpretations of probability” largely took the form of pitched battles between frequentists and related schools (who held that probability dealt with relative frequencies or objective propensities) and subjective Bayesians (who took probability to be about degrees of belief, subject to some constraints). But in recent years, a more objective and logical Bayesian interpretation has gradually come to the fore. Statisticians felt the need for some objectivity in prior probabilities, which permitted solid results in a great range of applied areas such as image processing. Legal theorists similarly felt the need for an objective understanding of uncertainty in legal decision making, and there has been extended debate about the use of Bayesian methods in legal cases involving DNA and other identification evidence. Bayesian networks have become a popular method of representing knowledge and making causal inferences in artificial intelligence. Philosophers added objective logical theories to the range of options they considered. It was particularly noticed that probabilistic reasoning works with the confirmation of conjectures in pure mathematics, where there are only logical relations, implying that there must be a purely logical interpretation of probability applicable in those cases. Objective Bayesian approaches to the philosophy of science would seem to be warranted, but have been less popular.

Public understanding of the Bayesian perspective was advanced by Sharon Bertsch McGrayne’s semipopular 2011 history, The Theory That Would Not Die19. (Franklin, “Preface to the 2015 Edition”—pages unnumbered)

As always, there is much more to delve into in this book. For example, its concluding chapter features a section called “Kinds of Probability and the Stages in Discovering Them,” in which—after noting that the book is organized around the distinction between factual and logical (i.e., what I’ve come to call “aleatory” and “epistemic”) probability—he discusses the “four factors that confuse the issue further, making it more difficult to decide what to look for in early authors” (Franklin 326). These are, to summarize:

First, actual people’s partial belief in, or uncertainty or doubt about, propositions and the support they give one another (sometimes called subjective probability) has some relation to logical probability but is not the same as it; indeed, psychological experiments show consistent discrepancies between the two, though the deviations are not too large in normal circumstances. …

Second, actual relative frequencies, or proportions in populations, have some relation to factual probability but are not identical to it. …

Third, there are connections between logical and factual probability, in that if the outcome of an experiment has a factual probability other than 0 or 1, the experimenter is rationally uncertain of the outcome. …

Fourth, the mathematical theory of probability can be, and usually has been, developed without reference to what kind of probability is being spoken of. (Franklin 326)

The confusions are, by now, not so surprising, though his explicit mention of them is refreshing. It’s a fascinating book overall, that covers far more than I’ve touched on here or in the other post I recently mentioned it in: “Defense Attorney’s Fallacy: A Conditional Probability Problem.” I also note there some points of tension between this book and Hacking’s.

This seems like a good place to take stock of our concepts and terminology so far, looking briefly at three more authors.

To be clear, factual or aleatory probability does not mean truly or fundamentally probabilistic, but, rather, that there is some statistically verifiable (whatever that means) probability. It is, as Franklin (and others) put it, “essentially numerical” (Franklin 326). An example of aleatory probability would be the rolling of a die or the rates of car accidents among 17-year-old American males. An example of epistemic or subjective probability, on the other hand, may or may not make sense to express as numbers—e.g., the probability that God exists or, perhaps, something that seems to accommodate numbers but is complex, like the probability we assign to whether Trump will get reelected: there are simply to many complex, moving parts—many of them coming down to inscrutable human psychology (despite how obvious Trump’s winning the first time may appear in retrospect; that’s an illusion)—even though in principle, God would be able to predict with 100% accuracy the outcome of Trump’s reelection; well, unless it turns out human brain function is ontically probabilistic, in which case God would make a 100% accurate and precise prediction that Trump wins with some probability between 0 and 1 (with who knows how many decimal places).

We may want “ontic” to mean the same thing for randomness, in which case a coin flip would not be truly or fundamentally random; or we might like to use the word differently in that context, so that a coin flip really is random, even though it’s probability is aleatory. I prefer to use the word “ontic” here, but will revisit the question in the next section, when I consider Chakraborty’s paper, where randomness per se is central.

The authors I’ve discussed so far do not discuss randomness with such subtly, as distinct from probability. I am familiar with some texts that do; e.g., Robert von Mises defines “randomness” in his 1957 essay, “Definition of Probability,” 20 but this is in service of spelling out a frequentist view of probability and doesn’t address the subtler questions of randomness that Chakraborty is motivated to address as a computer scientist.

3.5 Superforecasting: The Art and Science of Prediction by Philip E. Tetlock & Dan Gardner (2015)

Here, Tetlock & Gardner (T&G) discuss epistemic and aleatory uncertainty. Tetlock, who is on faculty at Wharton, is a psychologist who seems to have at least one foot in political science and decision making and such, and this perspective seems to inform the discussion. (Recall that definitions for these terms were also cited above from the Wikipedia entry Uncertainty Quantification, where the word “aleatoric” is used instead of “aleatory.”):

An awareness of irreducible uncertainty is the core of probabilistic thinking, but it’s a tricky thing to measure. To do that, we took advantage of a distinction that philosophers have proposed between “epistemic” and “aleatory” uncertainty. Epistemic uncertainty is something you don’t know but is, at least in theory, knowable. if you wanted to predict the workings of a mystery machine, skilled engineers could, in theory, pry it open and figure it out. Mastering mechanisms is a prototypical clocklike forecasting challenge. Aleatory uncertainty is something you not only don’t know; it is unknowable. No matter how much you want to know whether it will rain in Philadelphia one year from now, no matter how many great meteorologists you consult, you can’t outguess the seasonal averages.

You are dealing with an intractably cloud-like problem, with uncertainty that it is impossible, even in theory, to eliminate. Aleatory uncertainty ensures life will always have surprises, regardless of how carefully we plan. Superforecasters grasp this deep truth better than most. (T&G, 143–144)

This seems to be a different use of “aleatory” than I’ve settled on. The Wikipedia entry used the example of “random and complicated vibrations of the arrow shaft” in arrows with otherwise identical shooting setups to illustrate aleatory uncertainty. It seems to me that, in principle, those things are indeed knowable. The Wikipedia entry seems to admit this by the end of that definition; again: “Just because we cannot measure sufficiently with our currently available measurement devices does not preclude necessarily the existence of such information, which would move this uncertainty into the below category.”

And elsewhere, we’ve even seen the examples such as die rolls and coin flips used to illustrate aleatory uncertainty (examples aren’t hard to find; here’s another I quickly found via Google).

Now, I agree that, in practice, such a thing may be hopelessly out of reach, and for reasons I’ve already noted. This could just come down to what we mean by “in principle” or “in theory.” In principle, anything that is deterministic could be predicted given enough information and computing power—such as is possessed by an omniscient God. But this doesn’t seem to be quite what T&G mean. Nor are they using “aleatory,” I don’t think, to say what I mean by “ontic.” They seem to mean something more practical.

Consider, for example, the following notable distinction. The uncertainty of a vigorously flipped coin in an interestingly stochastic environment (e.g., landing on surface where the coin will bounce and roll around before resting) is irreducible and thus aleatory in the sense that you simply assign 50-50 until you see what happens. T&G use the more complex example of a currency-market question (T&G 144). While the uncertainty surrounding, say, whether the project scheduled to start next month will be finished by its six-month deadline is reducible and thus epistemic given the knowledge gained as the deadline approaches. This latter example may be closer to the aforementioned arrow example, as the deadline prediction may go from aleatory to epistemic, provided it becomes knowable “at least in theory.”

At any rate, I seem to only be able to find this sort of distinction between aleatory and epistemic uncertainty in discussions by people concerned with risk management and assessment in fields like engineering, medicine, project management. While philosophers and probability theorists seem to use “aleatory” to mean something more like the “true” probability of an event (e.g., on frequentist grounds) while epistemic is used to essentially mean “subjective,” i.e., a probability assigned given what you know, whether or not it is perfectly knowable “at least in theory.” The “subjective” interpretation of “epistemic” probability is particularly consistent among philosophers and probability theorists, though see again the points made the likes of Hacking and Franklin about distinctions even there.

A takeaway here, then, is that it’s possible for two people to be talking about epistemic probability while meaning very different things. The key is to be as clear as we can in our usage.

T&G does discuss at length a common variation on the idea of subjective probability—i.e., Bayesian probability (see Chapter 8, “Perpetual Beta” for more on that). Tetlock’s rigorous research into the application of Bayesian reasoning is important21. The book is worth reading for that alone. It’s excellent to see Tetlock working with, and contributing to the development of, forecasters who are exercising the translation of their probabilistic intuitions about highly complex—often socially driven—events into numerical assignments; often with great results (thus the label of “superforecaster”). There’s lots else to recommend here as well. I found his adaptation of the fox-hedgehog distinction inspired, especially the image of a dragonfly-eyed fox. And there is a fascinating, and instructive, discussion of Sherman Kent’s work at the CIA; such as, in 1951, when Kent realized that his team meant the phrase “serious possibility” to represent anywhere from 20% to 80% likelihood (!) as used in an official report the team signed off on regarding the potential for a Soviet attack on Yugoslavia that year (T&G 55).

The idea of probability applications to the Real World leads me to another recommendable book that should be known to anyone interested in discussions of probability past, present, future.

3.6The Black Swan: The Impact of the Highly Improbable (Second Edition) by Nassim Taleb (2007/2010)

I love this book. Taleb (whose academic training is in business but with a heavy quant emphasis and who has worked, among other things, as a successful hedge fund manager) proudly writes as a non-academic, generalist philosopher-scholar-essayist possessed of technical acumen and sensitivity to the Real World—along the lines of the personal hero-mentors he lauds in the book, such as Henri Poincaré, Michel Eyquem de Montaigne, Marcus Tullius Cicero, and, perhaps especially, Yogi Berra and one of Taleb’s own collaborator, the great iconoclast Benoît Mandelbrot. Make sure you get the Second Edition, even if just for its excellent, and essential, “Postscript Essay,” written three years after the book’s original publication.

I admit, Taleb sometimes comes off as a bit, how to put it? Smug? Arrogant? Self-Impressed? And sometimes mean. Too mean. I’m not a fan of mean. But the combination of humor and intellectual brilliantly outshine the the bad here. And the “Postscript Essay” mitigates some of this—e.g., he seems to demonstrate a more appropriately appreciative attitude towards academic philosophers. On the other hand, this is where he introduces to the book the unfortunate, cringe-inducing term Asperger Probability (though the idea it represents is interesting; see below).

Taleb—who, interestingly, has heaped praise on Franklin’s The Science of Conjecture in its (as of today) only Amazon customer review; while Franklin, in his book, characterizes The Black Swan as “the most successful of several works that explained [in the 2008-market-crash era] an inherent difficulty in predicting rare events”(Franklin 2015 Preface)—writes not just of probability but also of randomness per se, and recognizes the sorts of distinctions I’ve been on the lookout for. “Recognizes” might be the wrong term. “Calls out,” maybe. I’ll begin with a passage that might just as well have been written in response to this blog post. From the “Postscript Essay”:

… assuming that “randomness” is not epistemic and subjective, or making a big deal about the distinction between “ontological randomness” and “epistemic randomness,” implies some scientific autism, that desire to systematize, and a fundamental lack of understanding of randomness itself. It assumes that an observer can reach omniscience and can compute odds with perfect realism and without violating consistency rules. What is left becomes “randomness,” or something by another name that arises from aleatory forces that cannot be reduced by knowledge and analysis.

There is an angle worth exploring: why on earth do adults accept these Soviet-Harvard-style top-down methods without laughing, and actually go to build policies in Washington based on them, against the record, except perhaps to make readers of history laugh at them and diagnose new psychiatric conditions? And, likewise, why do we default to the assumption that events are experienced by people in the same manner? Why did we ever take notions of “objective” probability seriously? (Taleb Kindle Locations 6371–6378)

Fair enough. I’ve expressed plenty of concern myself about what it means for there to be a true, or aleatoric, probability for an event, given that this will come down to a model that it to some degree arbitrary (see again the deteriorating coin example). As for ontic probability, I’m not totally convinced it exists or that; and if it does exist, I’m not convinced it matter for anything but highly arcane contexts. Increasingly, it seems all probability is epistemic in the end. And, as Taleb points out here and Skyrms point out above and plenty of others are quick to point out: we aren’t all operating with the same beliefs, knowledge, experience (even when looking at the same evidence). Thus the connection—often synonymously drawn—between epistemic and subjective probability.

That in mind, here are some illuminating excerpts:

Probability can be degrees of belief, what one uses to make a bet, or something more physical associated with true randomness (called “ontic,” on which later).(Taleb Loc 5891–5893). …


I will briefly state some of the difficulties in understanding the message and the ideas of this book, typically perpetrated by professionals, though, surprisingly, less by the casual reader, the amateur, my friend. … (Taleb Loc 6090–6092). …

11) Spending energy on the difference between ontic and epistemic randomness—true randomness, and randomness that arises from incomplete information—instead of focusing on the more consequential difference between Mediocristan and Extremistan. (People with no hobby, no personal problems, no love, and too much free time.) (Taleb Loc 6117–6120; from a list of “Main Errors in Understanding the Message”). …


There is another difference here, between “true” randomness (say the equivalent of God throwing a die) and randomness that results from what I call epistemic limitations, that is, lack of knowledge. What is called ontological (or ontic) uncertainty, as opposed to epistemic, is the type of randomness where the future is not implied by the past (or not even implied by anything). It is created every minute by the complexity of our actions, which makes the uncertainty much more fundamental than the epistemic one coming from imperfections in knowledge.

It means that there is no such thing as a long run for such systems, called “nonergodic” systems—as opposed to the “ergodic” ones. In an ergodic system, the probabilities of what may happen in the long run are not impacted by events that may take place, say, next year. Someone playing roulette in the casino can become very rich, but, if he keeps playing, given that the house has an advantage, he will eventually go bust. Someone rather unskilled will eventually fail. So ergodic systems are invariant, on average, to paths, taken in the intermediate term—what researchers call absence of path dependency. A nonergodic system has no real long-term properties—it is prone to path dependency.

I believe that the distinction between epistemic and ontic uncertainty is important philosophically, but entirely irrelevant in the real world. Epistemic uncertainty is so hard to disentangle from the more fundamental one. This is the case of a “distinction without a difference” that (unlike the ones mentioned earlier) can mislead because it distracts from the real problems: practitioners make a big deal out of it instead of focusing on epistemic constraints. Recall that skepticism is costly, and should be available when needed. (Taleb Loc 6329–6342).

Taleb seems to be using the words “ontic” and “ontological” roughly as I’ve used them, with perhaps more than a little bit of the “aleatory” concept seeping in there (he uses the latter term once in the book, in the first passage I cited from the book). But I won’t pick nits here, as his central point seems to be that, whatever philosophical distinctions we make, epistemic limitations bind us. In other words, it seems that two broad distinctions work for whatever events humans are addressing: the probabilities the assign right now, and the probabilities they assign as they get more information. The latter is better insomuch as it improves one’s decisions in the face of uncertainty. (Where there is certainty—in particular, warranted certainty—there is no need for probability.)

(Perhaps I should also be looking at his 2004 book Fooled by Randomness, but I have not read that one, nor do I own it. A search at Amazon preview shows that “aleatory” doesn’t come up in that book at all. I’ll resist the urge to dig around there further.)


Taleb uses words like “stochastic” and “random” in the usual, standard-dictionary ways, so I’ll not share those passages here (though they’re worth reading, as is the entire book!). More relevant is his note on Bayesian probability:

Interestingly, the famous paper by Reverend Bayes that led to what we call Bayesian inference did not give us “probability” but expectation (expected average). Statisticians had difficulties with the concept so extracted probability from payoff. Unfortunately, this reduction led to the reification of the concept of probability, its adherents forgetting that probability is not natural in real life. (Taleb Loc 6637–6641).

This is interesting interesting for a couple of reasons. First, compare this to the aforementioned notion of quantum (what I call “ontic”) randomness, in which nature is inherently probabilistic. Again, there may good reasons to be skeptical about that interpretations of quantum theory. But even if that interpretation is correct, there’s still something to the idea of many notions of probability existing only as a kind of discursive practice for dealing with uncertainty, and not anything “natural in real life.” For example, whether the defendant is guilty (what is the “true” probability in such cases?) or whether the coin I flipped five minutes ago landed Heads (there may be a “true” probability here: if the coin lands Heads 50% of the time, then we can reframe the question as the probability that you correctly guess how the coin landed, which is 50%; though, again, this assumes a frequentist standard).

Second, I get the impression that Taleb does not go in for the current tendency (in my experience) to use “subjective” and “Bayesian” probability synonymously. (To be clear, this need not conflict with the idea that Bayesian implies subjective; it’s just that the implication may not work the other way around. Sort of like saying all pears are fruit, but not all fruits are pears.) Taleb isn’t outright critical of Bayesian probability (as we’ll see below, he endorses the next book I’ll consider on the grounds of its Bayesian approach). But he is critical of a certain view of Bayesian probability that, perhaps ironically, holds up the approach as a means of eliminating subjectivity:

Another deficiency I need to point out concerns a strangely unrealistic and unrigorous research tradition in social science, “rational expectations,” in which observers are shown to rationally converge on the same inference when supplied with the same data, even if their initial hypotheses were markedly different (by a mechanism of updating called Bayesian inference). Why unrigorous? Because one needs a very quick check to see that people do not converge to the same opinions in reality. This is partly, as we saw in Chapter 6, because of psychological distortions such as the confirmation bias, which cause divergent interpretation of the data. But there is a mathematical reason why people do not converge to the same opinion: if you are using a probability distribution from Extremistan, and I am using a distribution from Mediocristan (or a different one from Extremistan), then we will never converge, simply because if you suppose Extremistan you do not update (or change your mind) that quickly. For instance, if you assume Mediocristan and do not witness Black Swans, you may eventually rule them out. Not if you assume we are in Extremistan.

To conclude, assuming that “randomness” is not epistemic and subjective, or making a big deal about the distinction between “ontological randomness” and “epistemic randomness,” implies some scientific autism… (Taleb Loc 6363)

Notice the overlap with an above-shared passage. Also, I won’t get into what he means by two of the most important ideas in the book: Mediocristan and Extremistan (read the book for that).

Those observations align with one of my central philosophical motivations: the separation of human minds, the fact that two people with similar backgrounds and educations—in fact, it can be the same person at different times on the same day!—looking at the same evidence can rationally arrive at different conclusions and maybe even will each think the other either out of their minds or disingenuous (why?!?), and so on. “Similar,” of course, doesn’t mean identical; small differences seem to matter: no two people have the same background and experiences and sensitivities and umwelt, etc.

There are so many examples of this in academia that I hesitate to start listing, but I will list a clear case, which I view as falling in the intersection of the fringes of a few contentious areas, including consciousness, computer science, and physics: Nick Bostrom’s argument that one of three propositions must be true, the most striking of which is that our world is, and thus you are, a computer simulation. The recommendation is that—after carefully considering the argument!—this should earn a subjective probability assignment of abut 33%, but it’s subjective probability, so it’s between you and your intuitions. I wrote about that here (I wrote about it several years ago, and won’t review it again now, but I imagine today’s me would order some revisions): “You Are (Probably Not) a Computer Simulation.”

At any rate, the role of subjectivity is worth dwelling on in the present context, which Taleb does (though I’ll take issue shortly with a certain aspect of how he characterizes the phenomenon).

Namely, the term “subjective probability” is elaborated on and given importance in the book, appearing even in section and subsection titles. Here’s one such instance, in a subsection called “Probability Has to Be Subjective”:

The notion that two people can have two different views of the world, then express them as different probabilities remained foreign to the research. So it took a while for scientific researchers to accept the non-Asperger notion that different people can, while being rational, assign different probabilities to different future states of the world. This is called “subjective probability.”

Subjective probability was formulated by Frank Plumpton Ramsey in 1925 and Bruno de Finetti in 1937. The take on probability by these two intellectual giants is that it can be represented as a quantification of the degree of belief (you set a number between 0 and 1 that corresponds to the strength of your belief in the occurrence of a given event), subjective to the observer, who expresses it as rationally as he wishes under some constraints. These constraints of consistency in decision making are obvious: you cannot bet there is a 60 percent chance of snow tomorrow and a 50 percent chance that there will be no snow. The agent needs to avoid violating something called the Dutch book constraint: that is, you cannot express your probabilities inconsistently by engaging in a series of bets that lock in a certain loss, for example, by acting as if the probabilities of separable contingencies can add up to more than 100 percent.

There is another difference here, between “true” randomness (say the equivalent of God throwing a die) and randomness that results from what I call epistemic limitations, that is, lack of knowledge. (Taleb Loc 6320–6331).

Note again that this overlaps with an earlier cited passage. Also note that his use of the term “Asberger’s” is an adaptation of the idea that those with Asberger’s possess an “underdevelopment of a human faculty called ‘theory of mind’ or ‘folk psychology’” (Taleb, Kindle 6264). He goes so far as to name a chapter “Asberger and the Ontological Black Swan” (Chapter IV), a section of which is titled “Asberger Probability,” which is what it sounds like.

I’m uncomfortable with this terminology, which could be replaced with something that doesn’t play on an “atypical” cognitive style that may amount to something like a different kind of theory of mind, rather than something to be viewed as a disability in absolute terms (i.e., in contrast with manifesting as a disability because it’s made to be a hindrance by social attitudes and structures etc., and which perhaps should not be so casually generalized across the autism spectrum in the first place).

I’m not extensively educated on this subject. To hear a great interview with someone who is, listen to or read the transcript of this interview with Michelle Dawson on Tyler Cowen’s podcast: Michelle Dawson on Autism and Atypicality (Ep. 46).

That in mind, I’ll suggest a term that does an even better job of getting the same idea across, and without the insensitive, perhaps even harmful, associations: Egocentric Probability. Egocentricism is essentially the inability to distinguish your mental state from mine: if we are both looking at a model of a house, and I’m looking at the back and you the front, I, as someone who is paradigmatically egocentric, think you see the front—i.e., I think you see what I see. To think this way—to such an extreme—as an adult would be to thoroughly lack theory of mind (sometimes called “theory theory,” but I’ve read books and papers and had professors who used these terms, and yet other potential synonyms, differently; unsurprisingly).

We are all prone to egocentrism to some degree or another, and on some days and given some kinds of situations more than others etc., but in its most extreme forms it is associated with very young children who have yet to develop a robust theory of mind. I’m by no means up on the state of that research, either, but I can share the sorts of videos you tend to encounter on this topic when you get to the cognitive development section of a Psych 101 course; at YouTube: Egocentrism. There may be other, or additional and overlapping, explanations for why the child answers as the child does in that video, and I doubt the child completely lacks theory of mind, but I won’t get into these concerns here.

The point is that there’s an idealized, textbook notion of egocentrism, and this better accomplishes what Taleb aims to get across, which is to drive home the degree to which subjectivity permeates probability (without detection). Recall again an example I used above: if I flip a coin and conceal the result from you, the probability that it landed Heads is either 0 or 1 for me, but still 1/2 for you. Were I paradigmatically egocentric, I’d assume that you know just as I do how the coin landed. Taleb is right that this sort of mistake is a persistent problem, though of course more commonly so in more complex situations than this, or, more generally, in the assumption that there is some sense in which events should have the same subjective probability for all minds (which gets us into a discussion of Rationality, which I won’t touch here; it will be invoked again below, when considering Jaynes). Giving this problem a name might be a good idea. I nominate Egocentric Probability. (Compare this with Jaynes’s mind projection fallacy, of which egocentric probability might be a special case.)

While autism has in the past been, and often still is, characterized as a kind of extreme egocentrism disorder—see, for example, this release from 2009 and this undated entry at the Autism Research Institute on “Theory of Mind”—it strikes me that there is no necessary connection between those concepts as such and, again, this has been brought into question as the autism spectrum has been more responsibly researched. And even if it were, I don’t endorse what seems clearly meant as a negative application of the term by Taleb.

Finally, I’ll triply reemphasize that I’m far from expert on this subject, though I do hope I’m appropriately confused about it (a state I aspire towards in many domains): I don’t know what aspects of autism it helps or hurts those on the spectrum to say are disabilities in some important sense versus what aspects that are only made “disabilities” due to current attitudes and social structures, and those aspects that should (in some principled sense of “should”) be viewed as atypical mainly insomuch as they contribute (healthily or neutrally) to cognitive diversity, or are simply different in a sense of being on the far side of variance, but no more so than many cognitive styles that wouldn’t be (currently) “diagnosable” or considered, in any official way, “atypical,” and so on and so forth. And so again I highly recommend the aforementioned Michelle Dawson interview, as she is someone who can and does address such issues (notice, for example, her point that some folks seem much better off simply not being diagnosed, thus avoiding being treated as suffering a certain category of “disability”).


I’ll close the survey of Taleb’s book with one more excerpt. I of course haven’t done justice to the book’s scope or depth. You’ll need to read it for that.

This excerpt addresses the closest thing I could find in the book to discussing what I mean by “ontic” probability; note the emphatic clarity of his dismissal:

The greater uncertainty principle states that in quantum physics, one cannot measure certain pairs of values (with arbitrary precision), such as the position and momentum of particles. You will hit a lower bound of measurement: what you gain in the precision of one, you lose in the other. So there is an incompressible uncertainty that, in theory, will defy science and forever remain an uncertainty. This minimum uncertainty was discovered by Werner Heisenberg in 1927. I find it ludicrous to present the uncertainty principle as having anything to do with uncertainty. Why? First, this uncertainty is Gaussian. On average, it will disappear—recall that no one person’s weight will significantly change the total weight of a thousand people. We may always remain uncertain about the future positions of small particles, but these uncertainties are very small and very numerous, and they average out—for Pluto’s sake, they average out! They obey the law of large numbers we discussed in Chapter 15. Most other types of randomness do not average out! If there is one thing on this planet that is not so uncertain, it is the behavior of a collection of subatomic particles! Why? Because, as I have said earlier, when you look at an object, composed of a collection of particles, the fluctuations of the particles tend to balance out. (Taleb Loc 5493–9002)

[Note: Regarding averaging out, see Taleb’s comments about how the particles composing his coffee cup don’t randomly move in such a way that makes his cup spontaneously hop two feet into the air. (Taleb Loc 4625–4630)]

But political, social, and weather events do not have this handy property, and we patently cannot predict them, so when you hear “experts” presenting the problems of uncertainty in terms of subatomic particles, odds are that the expert is a phony. As a matter of fact, this may be the best way to spot a phony.

I often hear people say, “Of course there are limits to our knowledge,” then invoke the greater uncertainty principle as they try to explain that “we cannot model everything”—I have heard such types as the economist Myron Scholes say this at conferences. But I am sitting here in New York, in August 2006, trying to go to my ancestral village of Amioun, Lebanon. Beirut’s airport is closed owing to the conflict between Israel and the Shiite militia Hezbollah. There is no published airline schedule that will inform me when the war will end, if it ends. I can’t figure out if my house will be standing, if Amioun will still be on the map—recall that the family house was destroyed once before. I can’t figure out whether the war is going to degenerate into something even more severe. Looking into the outcome of the war, with all my relatives, friends, and property exposed to it, I face true limits of knowledge. Can someone explain to me why I should care about subatomic particles that, anyway, converge to a Gaussian? People can’t predict how long they will be happy with recently acquired objects, how long their marriages will last, how their new jobs will turn out, yet it’s subatomic particles that they cite as “limits of prediction.” They’re ignoring a mammoth standing in front of them in favor of matter even a microscope would not allow them to see. (Taleb Loc 5492–5513)

3.7 Probability Theory: The Logic of Science by E.T. Jaynes (2003)

The final book I’ll dip into is one that (picky!) Taleb recommends as “the only mathematical book other than de Finetti’s work that I can recommend to the general reader, owing to his Bayesian approach and his allergy for the formalism of the idiot savant.” (Taleb Loc 7390–7392): E.T. Jaynes’ dense, 2003 Probability Theory: The Logic of Science. I haven’t yet read the whole thing yet but its definitely on my short stack of in-progress reads and I’m excited to survey it today.

This book is a monumental achievement and is made heavy by both the weight of its ideas and its graduate-level mathematics—as the back cover puts it, it applies new mathematical results and probability theory to a “wide variety of problems in physics, mathematics, economics, chemistry, and biology” and “contains many exercises and problems, and is suitable for use as a textbook on graduate level courses involving data analysis.” In other words, it comes with more prereq’s than the other books I’ve looked at here. I recall once seeing a review by a reader who said you should be awarded a PhD just for getting through it.

The Wikipedia entry on Edwin Thompson Jaynes notes that the book

… gathers various threads of modern thinking about Bayesian probability and statistical inference, develops the notion of probability theory as extended logic, and contrasts the advantages of Bayesian techniques with the results of other approaches. This book was published posthumously in 2003 (from an incomplete manuscript that was edited by Larry Bretthorst).

Here are some relevant illuminating excerpts:

…it is seldom noted that [the verb “is”] has two entirely different meanings. A person whose native language is English may require some effort to see the different meanings in the statements: “The room is noisy” and “There is noise in the room.” But in Turkish these meanings are rendered by different words, which makes the distinction so clear that a visitor who uses the wrong word will not be understood. The latter statement is ontological, asserting the physical existence of something, while the former is epistemological, expressing only the speaker’s personal perception.

Common language—or, at least, the English language—has an almost universal tendency to disguise epistemological statements by putting them into a grammatical form which suggests to the unwary an ontological statement. A major source of error in current probability theory arises from an unthinking failure to perceive this. To interpret the first kind of statement in the ontological sense is to assert that one’s own private thoughts and sensations are realities existing externally in Nature. We call this the “mind projection fallacy,” and note the trouble it causes many times in what follows. But this trouble is hardly confined to probability theory; as soon as it is pointed out, it becomes evident that much of the discourse of philosophers and Gestalt psychologists, and the attempts of physicists to explain quantum theory, are reduced to nonsense by the author falling repeatedly into the mind projection fallacy. (Jaynes 21–22)

Jaynes’ mind projection fallacy (which has its own little Wikipedia entry) is intimately related to my aforementioned “central philosophical motivation” of the separation of minds. Namely, we bridge that mind gap by way of representation—or models—of the minds (and personal identities, etc.) of others. This extends to the external world in general, either as straight perceptual representation (e.g., of the objects in the room around you right now) or metaphysically (e.g., notions about identity over time). This even includes representations of ourselves, including representations of the models of ourselves contained in the minds of others!

Somewhere bound up in all this is our representations of the world—past, present, future—as a place of uncertainty: i.e., as probabilistic models, which sometimes fall under formal mathematical representation, sometimes under something more qualitative or intuitive (also perhaps with some degree of formalism). These notions of representation, which also include the whole array of the aforementioned ontological questions (what is a tree, a social group?), are what seem to tie together the whole range of my philosophical interests.


From a subsection titled, “‘Subjective’ vs. ‘Objective’”:

These words are abused so much in probability theory that we try to clarify our use of them. In the theory we are developing, any probability assignment is necessarily “subjective” in the sense that it describes only a state of knowledge, and not anything that could be measured in a physical experiment. Inevitably, someone will demand to know: “Whose state of knowledge?” The answer is always: “That of the robot—or of anyone else who is given the same information and reasons according to the desiderata used in our derivations in this chapter.”

Anyone who has the same information, but comes to a different conclusion than our robot, is necessarily violating one of those desiderata. While nobody has the authority to forbid such violations, it appears to us that a rational person, should he discover that he was violating one of them, would wish to revise his thinking (in any event, he would surely have difficulty in persuading anyone else, who was aware of that violation, to accept his conclusions).

Now, it was just the function of our interface desiderata (IIIb), (IIIc) to make these probability assignments completely “objective” in the sense that they are independent of the personality of the user. They are a means of describing (or, what is the same thing, of encoding) the information given in the statement of a problem, independently of whatever personal feelings (hopes, fears, value judgments, etc.) you or I might have about the propositions involved. It is “objectivity” in this sense that is needed for a scientifically respectable theory of inference. (Jaynes 43–44)

The robot he refers to is a brain “designed by us so that it reasons according to certain definite rules” (Jaynes 9). In other words, it establishes a transparent baseline of what we’d generally expect or hope for from a rational person. The “rules” are the desiderata referred to in the above passage. I won’t list these, but will note that they are clear and reasonable—for example (IIIb) essentially says that the robot will take all evidence into account rather than arbitrarily ignoring information on ideological grounds (Jaynes 17).

I’ll let Jaynes’ definitions speak for themselves.


Jaynes provides several comments on what I’ve called “ontic” probability, but he calls it “physical” probability, and addresses it, unsurprisingly, in the context of quantum physics. These passages definitely merit review, both because of the relevance to the present inquiry and because he was a physicist (keep in mind the earlier notes of skepticism that have been struck on this topic):

In his presentation at the Ninth Colston Symposium, [Karl] Popper (1957) 22 describes his propensity interpretation as ‘purely objective’ but avoids the expression ‘physical influence’. Instead, he would say that the probability for a particular face in tossing a die is not a physical property of the die (as [Harald] Cramér (1946)23 insisted), but rather is an objective property of the whole experimental arrangement, the die plus the method of tossing. Of course, that the result of the experiment depends on the entire arrangement and procedure is only a truism. It was stressed repeatedly by Niels Bohr in connection with quantum theory, but presumably no scientist from Galileo on has ever doubted it. However, unless Popper really meant ‘physical influence’, his interpretation would seem to be supernatural rather than objective. In a later article (Popper, 1959 26) rejects symmetry arguments, thereby putting his system of ‘personalistic’ probability in the position of recognizing the need for prior probabilities, but refusing to admit any formal principles for assigning them. (Jaynes 340)


In a more thorough survey, I’d share some of his excerpts with other words (a search in the Kindle edition revealed over 450 instances of “Bayesian,” not including in the index), and would perhaps venture out to dig around for responses to Jaynes’ critique of the Cophenhagen interpretation. But I’ll leave this here.

3.7 Joseph Hanna on “Ontic”

The last source I’ll look at is the one to which Chakraborty refers us for a discussion of ontic probability: Joseph Hanna’s 1982 paper, “Probabilistic Explanation and Probabilistic Causality”27.

In the paper, which was presented at a panel discussion, Hanna (a philosopher of science, I believe) makes references to papers by other theorists—to immediate interlocutors—to whom he is responding, as well as to earlier papers of his own. He states at the offset, for example, that his paper’s aim is to expand the scope of the planned discussion of probabilistic explanation to include the subject of probabilistic causality, in the service of arguing that “recent attempts to develop a theory of objective probabilistic causality are vulnerable to the same criticisms that in a previous paper I have advanced against the theories of objective probabilistic explanation proposed by Salmon and Fetzer” (Hanna 181). There are several other theorists mentioned as well—e.g., Paul Humphrey’s 1981 article “Aleatory Explanations” is noted to be a “one of the focal points” (Hanna 181) of the paper.

Rather than dive into the tempting rabbit hole of that greater, early 1980s discussion (which no do doubt expands out in all directions, and not just into the past and future; and, hey, there’s that word “causality” again), I’ll focus on Hanna’s own dealings with the phenomena and concepts relevant to the present survey. Hanna’s arguments may be too complicated and subtle to do justice to here (particularly when removed from the context of the greater discussion). In fact, this survey will at best serve as an unsatisfying teaser for the paper itself, as I will have to continually resist the temptation to dig deeper into its arguments and examples. If this frustrates you, read the paper. When it comes to the purposes of the survey, however, the gist is enough.

In the first part of the paper, he argues that “even if the macro world is in some respects irreducibly stochastic, there are, nevertheless, no genuine probabilistic explanations of any specific macro events” (Hanna 181). The central concern here seems to be one that I raised above, having to do with specific events being, in some important sense, poorly defined. That is, when we talk about the true probability of, say, a coin flip, we are talking about a reference class of “symmetrical, two-sided coins,” but not this particular coin, which may not be precisely fair, and in fact may still be poorly defined in the sense that flipped exactly this way in this environment it will land the same way each time. We then go a step further and say that even if a coin flip (as a macro event) is ontically random—“irreducibly stochastic”—the specific event is still poorly defined. How? I earlier noted that there’s the trouble of deciding, for example, whether you figure this out after so many coin flips (during which the coin may wear down) or based on precise measurement of the coin, and so on.

Another issue, which is what Hanna focuses on, I’ll put this way. Suppose the probability of a coin flip is ontic, so that God assigns some precise probability while you’re holding the coin, poised to flip it. That probability would be updated as you move the coin in your hand, as it launches, with each spin and bounces on the ground, etc. It is difficult to define what we mean by the event this coin flip (a token event of the general event type “coin flip”), particularly if we’re meant to put a numerical probability on it; and this exact set of conditions may never be repeated. The upshot is that we’re necessarily assigning epistemic probabilities when we speak of the probability of an event in any general way.

I believe this illustrates and aligns with Hanna’s observations and claims. At any rate, I’ll share a more interesting example from Hanna himself in a moment. First, here’s his expression of the concern. Note the use of “ontic” and “epistemic” etc.:

The central problem of probabilistic (or statistical) explanation, from Hempel’s original presentation of the Inductive-Statistical model right down to the present, has been to provide an objective account of the explanation of particular events. The epistemic (as distinct from ontic) character of the Requirement of Maximal Specificity was certainly one of the focal points of both Salmon’s and Fetzer’s critique of Hempel’s I-S model. From this perspective, the problem has been to convert objective generic explanations—covering all events in some reference class, population, or experimental arrangement—into objective explanations of the specific events comprising the population or of the specific outcomes of the experimental arrangement. The “stumbling block” is that a specific event can be referred to a variety of populations, no one of which is in any objective sense the population to which the specific event belongs, and the explanation that is given for the specific event—the probability attributed to it—depends upon which of these populations it is referred to. Or, to state the difficulty in propensity language, a specific event can be viewed as the outcome of a variety of experimental arrangements, no one of which is in any objective sense the experimental arrangement which produced the event, and the explanation that is given for the specific event—the propensity attributed to it—depends upon which of these experimental arrangements it is identified with. (Hanna 182)

It seems to me that, by “epistemic” he means the usual. By “ontic,” I at this point think he means roughly what I mean by “ontic,” which I equated above to his use of “irreducibly stochastic,” but this is not yet entirely clear. “Irreducibly stochastic” could mean something like “aleatory” in the aforementioned arrow case, or as Tetlock meant it. A “stochastic” process is generally thought to be one in which the features are too complex, or the tiny bits of random events too numerous, to keep track of—imagine calculating where billiard balls will go on a boat in rocky waters, or what the pyramid bumper at the end of a craps table introduces into a dice throw, or the vibrating arrow shaft.

In my experience, the word “stochastic” already implies practical irreducibility, such that to use the word “irreducible” would be redundant. Hanna’s use of the word “irreducible” above, particularly while also referring to the “macro world”28 so far makes me think he means “ontic” in the same way I do. We’ll see if he says anything more explicit. And it seems to me that his use of “objective” is what I mean by “true” or “aleatory” (yeah, this term is getting more and more “fraught”) probability, which may or may not be ontic. By this point, it seems to me that “objective” is the best word for this.

If it’s not already clear, I am in total agreement with Hanna (and others already noted above) that the notion of an objective probability assignment is a fantasy. Conducting this survey has, I think, made this objection in me stronger than ever.

Solutions have been proposed, but they come down to setting arbitrary standards for the sake of explanation. That’s fine and maybe even important to do, but it is not objective. Here’s a passage that refers to such proposals:

Thus, Salmon’s S-R model identifies the ontic probability of a specific (token) event with the limiting relative frequency of its generic type in a certain objectively characterized reference class, while Fetzer’s C-R model identifies the ontic probability of a specific event with its propensity in the experimental arrangement determined by the class of all causally relevant factors actually present in the specific case. Either way, the attempt has been to associate a unique, objective probability with each specific event—what Humphreys (1981) refers to as the correct probability of the event. (Hanna 182–183)

Here, the terms “ontic” and “objective” appear to be synonymous, but they could be considered distinct concepts, given that it’s possible that, depending on one’s model, an ontic probability will be distinct, and, if you calculate that number precisely, you’ve calculated that ontic event’s objective (or correct or, as he puts it elsewhere, true) probability. But I might be pushing it in claiming that this is what Hanna means. And, again, I’m going to resist the temptation of diving into the rabbit hole, looking at his usage in past papers, etc. It’s interesting enough that, at this point in the paper, he has so far taken the term for granted—or as clear from context—though his usage is, for an future outsider to that discussion (i.e., to me), arguably ambiguous.

Hanna next introduces an example borrowed from the above-cited Humphreys paper: a car sliding off the road as it rounds the bend. In doing so, he makes an ontic-epistemic distinction, as follows:

…although Salmon and Fetzer are right to search for a theory of scientific explanation that is objective—objective in the sense that the adequacy of a purported explanation of some specific event A depends upon the ontic (rather than epistemic) probability of A—that search is destined to fail. It is destined to fail because for any actual specific event this ontic probability (propensity, if you like) is either undefined or else, if defined, it equals one so that the event is not even a candidate for a probabilistic explanation. (Hanna 183)

Here Hanna seems to equate “ontic probability” and propensity; see the first passage above of his characterization of his critique in Fetzer’s “propensity language,” the alternative to which (that Hanna mentions) is Salmon’s “relative-frequency” language. Recall the definition of propensity (mentioned in the above Jaynes survey) at the SEP entry, Interpretations of Probability:

Like the frequency interpretations, propensity interpretations locate probability ‘in the world’ rather than in our heads or in logical abstractions. Probability is thought of as a physical propensity, or disposition, or tendency of a given type of physical situation to yield an outcome of a certain kind, or to yield a long run relative frequency of such an outcome. (Subsection 3.5)

Hanna goes on to note that “micro events of the sort associated with quantum phenomena” raise problems outside of his present concerns, he specifically intends

to include macro consequences of micro events in the account. For example, the “click” of a geiger counter following the radioactive emission of an alpha particle is such a macro event. The point of my criticism of the theories of probabilistic explanation can be stated as follows: even if quantum theory gives a correct account of the universe (in which case the macro world is also irreducibly stochastic in at least some respects), there are no genuine probabilistic explanations of any macro events—including the “click” of the geiger counter. (Hanna 183)

Here again he uses the term “irreducibly stochastic,” and this time does seem to mean it in the way I mean “ontic,” though I’m increasingly convinced that by “ontic” he just means “objective,” “true,” and so on—but then why does he go out of his way to clarify “objective” and not “ontic,” while seemingly reserving these terms for certain contexts (e.g., as when comparing ontic and epistemic probability); see below excerpts for more instances.

Hanna then sets ups some formalisms and terms and background conditions for the car example. This is far more than I need to get into here. The essential point is that as the “as the system [or event] evolves over time [the] probability also changes,” resulting in a problem of “temporal explanatory ambiguity” (Hanna 184). This problem of temporal indexing aligns with my aforementioned observations about the probability of a coin’s landing Heads changing as it gets closer to the end of the event, which we’ll call here an “absorbing” state.

The ambiguity is explained as follows:

Temporal ambiguity arises because if the world is irreducibly stochastic, then the specific events that occur in its history do not have unique probabilities (or propensities): rather, those probabilities must be temporally indexed. In the discrete case one can (at best) identify the probability of a specific event with its objective propensity relative to the immediately preceding (causally complete) state of the system. (Hanna 184)

I think I’ve given just enough a sense of Hanna’s discursive (i.e., semantic, technical, etc.) style (interesting how these vary across fields, time, individuals; that’s largely the point of the survey) to introduce the car example. Again, this isn’t meant to be a thorough examination of the example (which I consider to be a fantastic illustration of the problems it addresses), but rather will get into it only to the extent that the present survey requires.

He first introduces it as having three stages: Beginning State, Intermediate State, Absorbing State. During which the breaks may lock up or not, and the car may crash or not. Hanna’s diagram makes the situation clear:


There must be more to the story here, of course. Hanna asks us to suppose is that the driver panics when rounding the bend, thus jamming the breaks so that they locked. He then puts our survey terms to use (note that Laplace’s demon is a hypothetical mind that knows enough about any given state of the deterministic universe to predict everything that will happen thereafter, and retrodict everything that happened hitherto):

We will suppose further that the indicated discrete state system reflects the actual, objective empirical laws—in particular, that the system is irreducibly stochastic in the sense that there are no “hidden variables” whose inclusion in the system would change the given transition probabilities. In other words, the indicated system includes all the factors that were “causally relevant” in the specific situation.

Now consider this irreducibly stochastic system from the perspective of a Laplacean demon who knows the values of all the “causally relevant” variables and who is able, instantaneously, to perform all required calculations involving the universal or stochastic laws that characterize the system. Since the system is assumed to be irreducibly stochastic, it follows that even such an omniscient demon will be unable to predict its outcome with certainty. Furthermore—and this is the crucial point—the probability (propensity) which the demon would attribute to a given outcome will change over time. In the present example, the probability that the demon would attribute to accident A… (Hanna 185)

(Review the changing probabilities between States in the diagram to see where he’s going with this.) Now we’re getting somewhere! Laplace’s demon can only assign a probability, and, as with my aforementioned coin example, the probability changes as the state changes. (I wonder, given ontic randomness, whether the demon—or God—can make accurate counterfactual assignations, and thus carry a kind of table of every possible scenario, or if it must make calculations as they occur.) The obvious place to go from here is that the breakdown into just three states is arbitrary and so could have been a much greater number taking into account many factors, e.g., those involved in whether the man panics, some of which would precede the above-designated Beginning State:

The point is that if the world is irreducibly stochastic then specific events do not have unique, objective propensities; rather, their propensities evolve over time. This fact about stochastic systems poses a critical difficulty for any theory of probabilistic explanation which requires that a genuine explanation must cite the “true” or “correct” probability of a specific event. For, even if it is granted that the transition probabilities in the particular discrete state system are genuine single-case propensities, it does not follow that the absorbing state (the accident A) has a unique single-case propensity. (Hanna 185)

Hanna addresses some possible objections and goes on to further flesh out the above points about arbitrariness etc. while taking the idea to the point of the series of events being effectively (or even actually) continuous. Such a breakdown has interesting, but problematic, results (e.g., if you consider the state a microsecond preceding the absorption state to be the “ontic single-case propensity for the event” [Hanna 187], it is effectively just 1 for the event occurring, which is trivial and useless for probability, and will continue indefinitely as you move backwards, as any two adjacent micro-states will have the effectively the same ontic propensity):

Thus, whether or not the causal process is deterministic, it seems clear that a microsecond before the accident takes place (and here we must suppose that the specific event A is defined in a precise if somewhat arbitrary way) it was virtually certain to occur. (Hanna 187)

The upshot is that

the ontic propensity for the event either eventually reaches one or else it is undefined (because the temporal sequence of transition probabilities never reaches an upper bound). In either case, according to Fetzer’s C-R model, there will not be a genuine probabilistic explanation of the event. (Hanna 187)

He then seems to acknowledge that there may be some conceptual shift to consider as discrete cases near continuity—namely one from macro determinism to micro indeterminism. And here something very interesting happens that I recommend reading the paper to get a better look at. He stands by his analysis, with particular commitment to the discrete case in terms of what he can confidently argue, and concludes:

Thus, genuine indeterminism is not logically incompatible with “near” continuity at the macro level. The question of whether genuine indeterminism is compatible with complete continuity raises difficult conceptual and mathematical problems for which I do not see clear solutions.

That conclusion ends with a footnote (#8):

In his spoken commentary on my paper, Professor Salmon observed that there is, in fact, a rather straightforward answer to the question of whether “genuine indeterminism is compatible with complete continuity.” He presented a simple example demonstrating that compatibility. Salmon’s example illustrates an ambiguity in the notion of continuity (between physical continuity and probabilistic continuity) that I did not guard against with sufficient care in the body of my paper. Since the distinction is important to my argument, I have explored it more fully in an appendix. (Hanna 190)

The appendix example involves a resting particle that, at any point in time, may stay at rest or move, with probability 1/2. The idea is that it demonstrates “the compatibility of continuity and genuine indeterminism” (Hanna 191). But Hanna challenges this:

Now, Salmon is certainly correct in his claim that this example illustrates a physically continuous process that is irreducibly stochastic (indeterministic). However, if we view it as a probabilistic process, it is clearly discontinuous …

Thus, on the assumption that the actual trajectory is f1, the prob- ability of event E makes a discontinuous jump from 1/2 to 1 at the point t=to. So the process is not probabilistically continuous.

I’ll close the survey with Hanna’s concluding remarks in his paper:

The arguments advanced in this paper are aimed at those theories of probabilistic explanation and probabilistic causality which presuppose a standard event ontology. I have argued that despite the role that irreducibly stochastic processes evidently play in the evolution of at least some macro systems, the events that are the absorbing states of such systems are not stochastic in any significant sense. Thus, the specific event A can be given a non-trivial probabilistic explanation relative to the “causally complete” state of the world at various stages of the stochastic process leading to it, but A cannot be given a probabilistic explanation in any absolute sense. In an earlier article, I made this point by distinguishing between dynamic events (temporal segments of causal processes) and static events (the absorbing states of such processes). There I argued that the former, but not the latter, are amenable to objective probabilistic explanation. In certain respects, this emphasis upon the dynamic character of the events that are explananda of probabilistic explanations complements Salmon’s tentative proposal to ground theories of probabilistic explanation and probabilistic causality upon a process ontology rather than upon an event ontology.

3.8 Google’s Books NGram Viewer

If my shallow dips into the generously deep pool of probability-speak leaves you wanting, I invite you to play around with Google’s Books NGram Viewer (and feel free to share any interesting results). In a search just now, I found no hits for “ontic probability” (though “ontic” is popular) or for “ontological probability.” But “physical probability” did yield several hits, though these do not necessarily align with “ontic” (in my sense); the first hit was Hacking’s above-discussed book, where it comes up once, in reference to the meaning of the word “possibilité” in an 18th-century French text (denotes “something like physical probability” [Hacking 131]); the second hit was from a 2007 book called Labeling Genetically Modified Food: The Philosophical and Legal Debate (by Paul Weirich), in which “physical probability” is illustrated with a coin toss.

The term “quantum probability” also yields interesting results.

And of course there are the rabbit holes of Wikipedia and SEP, where you might find something new every time you look. Consider, for example, the SEP entry on Imprecise Probability, an approach some characterize as a “natural and intuitive way of overcoming some of the issues with orthodox precise probabilities.” It’s a fairly technical article, updated in February 2019, featuring yet more terminology, like “epistemic voluntarism.”

I could go on listing resources (one more: the meta-pathway between rabbit holes known as the Internet (née “Indiana”) Philosophy Ontology Project), but now must return, finally, to Chakraborty’s paper.

Chakraborty’s Epistemic and Ontic Randomness

Chakraborty, a statistician by training, discusses two kinds of “genuine” randomness: ontic and epistemic randomness. He defines those adjectives in the Abstract as

Ontic: What is actual (irrespective of what we know).

Epistemic: What we know (hence it relates to our knowledge of something).

Again, he’s that he’s applying these terms to randomness, not probability. It will be interesting to now see the extent to which these two domains may be conceived of separately, particular when modified with the adjectives “ontic” and “epistemic.” The survey was in large part meant to give a kind of backdrop for such reflections, so they aren’t happening in isolation.

To begin, he characterizes randomness as follows:

A careful thought explains that randomness does not quite mean lawlessness. All it means is that the experimental set up has an inherent variability which accounts for the unpredictability of the response and hence the “randomness” of the random experiment. Remember while defining a random experiment as an experiment in which the possible outcomes are known but which one will materialize cannot be predicted in advance even if the experiment is repeated under similar conditions, we merely said the conditions are “similar”. Did we say they are exhaustive? Sure enough there are many more conditions in the experimental set up (that affect the response) that we did not take into account. And we did not for we have no control over them! Herein lies the “inherent variability”. (Chakraborty 1)

As expected, he next identifies two sources of “inherent variability.”

The first is “precisely what the philosophers call ontic indeterminism or ontic randomness,” which he describes as “the law (i.e. the exact model used by nature) is changing from realization to realization and thus we are unable to describe the system’s behaviour by a fixed law” (Chakraborty 1). He gives an example:

a complex biological process such as the response of a person complaining of a certain set of symptoms to a certain treatment on different occasions. If the law were fixed on each occasion, medical science would be an easy profession! Again it does not mean that there is no law. In every realization there is a definite law that governs the cause-effect relationship. But the law holds for that realization only and changes in the next, the next and the next! Gentlemen, don’t you feel it is this dynamic nature of the law that we are misinterpreting as “lawlessness”? (Chakraborty 1–2)

This strikes me not as ontic as I’ve used the term, but as aleatory as with the arrow shaft case. In other words, where indeterminism—or what some call quantum randomness29—denies the strict picture that this exact cause results in that exact effect every time you rewind the tape of the universe to the instant the cause occurs, while aleatory randomness is essentially a matter of complexity. In the former case, the best answer—God’s answer —to what will happen? is some number between 0 and 1 exclusive (for if it were 0 or 1, then you would get the same result every time you rewind the tape), while in the latter case, God or Laplace’s demon would be able say with certainty (i.e., assign a correct probability of 0 or 1) to the outcome (e.g., by way of the “definite law that governs the cause-effect relationship”).

The inherent variability in the ontic case, for Chakraborty, seems to be the fact that no two cases are exactly alike. I agree that, in practice, this makes things like biological systems so complex that they might as well be ontic—it’s a good bet that the limits of practical intelligence, even that of the best brains possible (artificial or otherwise), are not equal to the limits of the world’s complexity. But conceptually or theoretically, there is a distinction between ontic and aleatory. And maybe it’s one worth thinking about, or at the very least worth keeping conceptually and semantically clear (particularly while things like the Copenhagen interpretation are floating around).

On the other hand, Chakraborty does use the word “indeterminism,” and he gives an example below that may be closer to my sense of “ontic,” involving a gas molecule. On the third hand, he will soon use the term “epistemic indeterminism”! More on these points in a moment, when we look at Chakraborty’s efforts to make clearer the ontic-epistemic distinction he is describing at this point in the paper.

Now compare Chakraborty’s characterization of ontic randomness with that of epistemic randomness—i.e., the second source of inherent variability: “the law is fixed but some or all of the parameters are not” (Chakraborty 1). The example he gives here is of a coin flip:

In contrast actual coin tossing, reflecting the second case, is a purely mechanical process and therefore can be described by Newton’s laws of motion30, and hence it cannot be random “in principle” (that is not in the ontic sense) but is still random “in practice” (that is random in the epistemic sense) for we certainly conceive it as random. Here the inherent variability is caused by the parameters some of which are clearly dynamic (parameters include the exact force applied, the point at which it is applied, the wind velocity, the parameters involved in the bombardment of the air molecules with the molecules of the coin when the coin is spinning in the air etc.……can you fix all of them in every toss?).

This strikes me as a reasonable account of epistemic randomness, and might help me better understand what he means by ontic randomness; however I find myself confused. He notes here that ontic randomness is randomness “in principle,” which maybe means something like what I mean by ontic (i.e., in the Copenhagen sense). But I’m not exactly sure how to gather this from his examples. True, a standard, vigorous (more on what I mean by “vigorous” shortly) coin flip is less complex than a biological system, but the principle seems to be the same.

In principle, if we knew all the parameters of this coin toss, we’d be able to predict the outcome with certainty. And the same goes, as far as I can tell, for, say, the medical treatment. I have further questions here, but will hold off until looking at Chakraborty’s clarifying comments, which are on the horizon.

First, it’s worth noting that we are talking here about a certain kind of coin flip. Namely what I call a vigorous flip. That is, one that isn’t somehow fixed so that the outcome is actually random (we might say, for example, so that the coin is free to express its bias, or even its propensity, etc.). Chakraborty addresses this criterion by bringing up the work of Persi Diaconis who, along with Susan Holmes and Richard Montgomery, whose 2007 paper, “Dynamical Bias in the Coin Toss,” begins with mention of a machine designed to flip a coin so that it lands heads 100% of the time, noting that they conclude from this that “coin tossing is ‘physics’ not ‘random.’”

This conclusion seems to imply that if something about coins made them truly random (what I call ontically random or probabilistic), no such machine could be made. You could set the coin up in the exact same conditions each time, and it would land Heads only some percentage of the time. But as things are, a coin in even just highly similar conditions will land the same way 100% of the time. This means that it’s not the coin itself, but that the environment also contributes to the randomness of a coin toss (notice that this may actual result in an ontically random coin flip; e.g., if the machine flips the coin one way or another, depending on the outcome of some other ontically random event). I’ve brought this up here already, as well as in other writings. I’ve also noted before that I take this idea to be implicitly understood in our concept of “coin toss”—such that the terms “coin toss” and “coin flip” etc. are understood to mean a non-fixed flip (e.g., you don’t just sort of let the coin plop out of your hand, flipping over once onto a nearby surface). Thus my use of the word “vigorous.” A vigorous coin toss is, by definition, one in which very few parameters are controlled; you get the idea.

Chakraborty’s response to Diacones largely aligns with my own, though it’ll be interesting to consider whether this illuminates Chakraborty’s notion of ontic randomness:

True, Persi Diaconis, the well known mathematician and former magician, could consistently obtain ten consecutive Heads in ten tosses of a coin by carefully controlling the initial velocity and the angular momentum (ref [1]) but the question of interest is: could he do this if the law were dynamic? Moreover it is not clear whether Diaconis could achieve it if the coin after falling over the ground rolled. Surely he must have tossed it over a rough surface (such as a magician’s table preferably with a cloth over it). I will not enter into this debate but it is clear that Diaconis was only attempting to fix the dynamic parameters. The law is already fixed whence it cannot be ontic indeterminism. We call it epistemic indeterminism in that it is not the system but the environmental factors contributing to the inherent variability in the experimental set up.

By environmental factors, we mean to say that there is a complex and unknown law (or even if known we are unable to write the exact model equation and solve it for a given set of parameters advance knowledge of all of which we do not have) which generates a sequence where we find hard to recognize any pattern (and hence random!), the dynamic parameters and our inability to smother this dynamism. But since this law is fixed it is not “in principle” contributing to the randomness howsoever complex it may be. It would have created the same response if only we could make the dynamic parameters static. It is to be understood that the inherent variability in the experimental set up is the causality and our ignorance/inability mentioned above a precipitating factor. (Chakraborty 2)

This would all be fine if he were comparing it to a kind of randomness described in the Copenhagen interpretation. It seems to me, however, that he is simply comparing it to a more complex event (e.g., medical treatment), which I would take to be aleatory. In fact, there’s a good argument to be made that vigorous coin flips are aleatory. How about a coin bouncing off of a thousand quivering arrow shafts?

Chakraborty makes an effort to clarify the ontic-epistemic distinction by way of a kind of classroom student-teacher Question & Answer dialog, and with some closing remarks.

The Q&A is addressed at “ontic indeterminism versus epistemic indeterminism.” There’s that term “epistemic indeterminism.” I would normally consider “indeterminism” to be another way of saying “ontically” or “irreducibly” or “inherently” probabilistic (i.e., in my usual sense of “ontic”). But I’ll interpret “epistemic indeterminism” as something like: full knowledge cannot be attained, at least in practice if not in principle. In other words, where “indeterminism” is another word for “randomness” or “inherent variability”—though this sounds more like “inherently random,” though I suppose on Chakraborty’s account “inherent randomness” would be “inherent inherent variability—rather than being the (Copenhagen-informed) opposite of “determinism” in the Newtonian sense Chakraborty uses it above. Phew.

That noted, how about the content of the Q&A? I’m tempted to reprint the whole thing here, but instead will pull out some interesting bits (the paper is easy to find online if you want to read the whole thing; here again is a link).

The first question asked is:

Q: What is common to ontic and epistemic indeterminism?

A: In both cases the law/laws are very complex and generate sequences where it is hard to find any pattern. In both cases we don’t know the exact law or are unable to write the model equation and solve it for a given set of parameters for a specific realization. (Chakraborty 2)

This answer helps in understanding the distinction between epistemic randomness and epistemic probability. That is, it seems ontic and epistemic randomness require epistemic probability. This helps in (my) understanding the nature of epistemic randomness—namely, surrounding the question of what sort of phenomenon it is. Is it just a name given to our epistemic and computational limitations? Or is it a phenomenon in the world, as it were, as ontic randomness is? On Chakraborty’s account, it must be. It deals in the Laplacian/Newtonian deterministic laws, which are only matters of probability under uncertainty. The question is whether this amounts to a kind of literal randomness, or if “randomness” is just the word we give to that probability-demanding uncertainty.

Keep in mind that Chakraborty considers epistemic randomness to be genuine randomness, unlike, for example, pseudo randomness. The distinction there is that pseudo randomness arises, say, from some algorithm (e.g., as in Excel), while epistemic randomness seems to emerge from the relation between human experience and Newtonian physics. In other words, for a human mind, pseudo-randomly generated numbers are no less random-appearing—no less predictable—than those generated by, say, vigorously rolling a die. So Chakraborty’s epistemic randomness is not just about epistemic limitations. It exists in the world as an emergent property of natural, deterministic processes and their phenomenological representation in the minds of conscious beings—particularly those who are trying to assess “What will happen next?

Why does this property not emerge in the case of pseudo randomness, given that the mind cannot tell the difference? The claim seems to be that this is simply a different sort of property—the mechanism is different, what exists in the world is different, there’s a principled way to give them different names (e.g., the pseudo randomness, which comes down to a bit of computer code, is not just theoretically predictable, but is predictable in practice provided the code is known).

This notion of emergence may be further than Chakraborty aims to take things, but it seems to me the way to go if we want to distinguish epistemic randomness from epistemic probability, and, for that matter, ontic and epistemic and pseudo randomness. It plausibly preserves epistemic randomness as a phenomenon in itself. Or maybe I am overthinking it and it’s just a word to give to the sort of randomness that we could, in theory (if not in practice), eliminate with the proper knowledge and computational power. That works too, I suppose.

In Chakraborty’s next answer, he succinctly spells out the main difference between the concepts as: “In ontic indeterminism, the law is also dynamic! In epistemic case, the law is fixed (e. g. Newton’s laws of motion in a coin toss)” (Chakraborty 2). I’m not sure I understand this notion of the law being dynamic, but it is strikingly ontic (in my meaning), rather than aleatory, language. I’m no expert in quantum mechanics, but my understanding in that regard is that the laws are fixed, it’s just that this happens to result in indeterminism—what I’ve heard referred to as the law being “chancy.” Even if the law were to change from trial to trial when trials are repeated, it seems that this change would be guided by some meta-law (i.e., this is the chancy law), which itself is indeterministic (in the usual sense of that term; not as Chakraborty seems to mean it, which is as “random” without qualification). Which, on Chakraborty’s view, may demand further regress to a meta-meta-law, and so on (rather than simply saying that the law does not change, so much, again, as it has randomness irreducibly built into it).

So, it seems to me the idea that the law is dynamic, in the sense of changing from one Newtonion-esque model to another, seems wrong to me. In other words, Chakraborty seems to characterizes the law responsible for each trial (or as he puts it, “realization”) as itself deterministic. And if it’s deterministic, the true probability, as it were, is 0 or 1. So the thing that’s really indeterminate (in principle) here is which deterministic model is in effect.

I believe this is roughly the correct interpretation of Chakraborty’s characterization of ontic; see again his above claim that the model used by nature changes from realization to realization, and that “in every realization there is a definite law that governs the cause-effect relationship” (Chakraborty 2). Such claims seem to come out of his interest in preserving the idea that randomness does not mean lawlessness, a mistake he seems to characterize as common, as he urges us to avoid it: “Gentlemen, don’t you feel it is this dynamic nature of the law that we are misinterpreting as ‘lawlessness’?” (Chakraborty 2).

Perhaps this is a push against the Copenhagen interpretation without quite moving over to the many-worlds (sometimes referred to as “Everettian,” after physicist Hugh Everett) interpretation (not, by the way, the difficulty of making sense of probability in the many-worlds case, given that it predicts with 100% accuracy that all the outcomes on the table will be realized). I’m not sure. And the example he gives of an ontic process doesn’t exactly clear things up:

Suppose for the sake of argument, a gas molecule is at position A at one moment and goes to position B at the next and to position C at the very next. How did it come to B from A unless there is some explanation-rule? Definitely there is a law howsoever complex it may be. The point of interest is: is the law that brought it from A to B the same as the law that brought it from B to C? If not, the indeterminism (which does not mean lawlessness) is clearly ontic. To the best of my knowledge, there is no fixed law that describes the movement of an individual gas molecule. Laws in Kinetic theory of gases are meant for a collection of gas molecules and not for an individual one!

The key point here is that the situation is ontic if the law that brought it from A to B is not the same as the law that brought it from B to C. This seems like “ontic” as I would describe it, though, again, I’m not sure what to make of the idea of a the law itself changing. Also note his invocation of complexity—perhaps this means each of these laws is, in itself, aleatory. Also, if each of these laws is essentially, in itself, something of the sort that, if fixed, would be describable in principle (as with Newtonian physics): do they cycle? Do they change randomly? Are there a finite number of them? And, again, is there some meta-law that determines these things? And so on.

At this point, I’ll simply rest on the basic interpretation of Chakraborty’s claim as: what comes out of the ontic law is indeterministic in principle, whatever the mechanism. Which is essentially what I mean by “ontic” as well.

He goes on to discuss (among other things) false randomness (pseudo and telescopic) which I won’t get into here.


This is the second time I’m writing this conclusion. The first time, I opened with the following sentence: “I will now conclude with an explanation of how I intend to use the terms at hand going forward, with the most clarity I can muster.” I started with the term “ontic randomness/probability,” and that went fine (I stuck to the use I’ve employed throughout these notes). But I immediately ran into problems of precision and clarity with other terms, including “epistemic” and “subjective” (I think Hacking might be right in avoiding the term “subjective,” but on the other hand, good luck doing that these days). Now that I think of it, while ontic randomness may be clear enough a concept, things get dicey when trying to nail down what a probability assignment will look like (including were God to give one).

So, instead, I’ll simply promise to try to be clear in context, with these notes as a reference and reminder that, if I look too closely, I’ll likely turn bewildered. As it should be.

In that spirit, I’ll include here my unfinished, un-edited first attempt to establishing a kind of glossary for future usage. I couldn’t muster the clarity.

First, I suppose I must ask myself to what degree I’ve satisfied Chakraborty’s to understand randomness (before endeavoring to understand probability). I might now be more confused about it. But maybe that’s good enough. Maybe you don’t have to understand randomness so much has to have thought very hard about it. Or at least that’s what I’ll tell myself as the more I think about it the less I understand it. Maybe that’s what’s to be understood: how hard it is. It’ll have to do, I guess.


Ontic randomness/probability: Randomness of the sort described in the Cophenhagen interpretation, which I’ve defined here too many times to repeat again. Probability that deals in such randomness. Synonyms are: irreducible randomness, truly random, inherently random, embedded randomness, ontological randomness, ontologically probabilistic, objectively random, and so on. Going forward, I will favor the clearest of these (e.g., “irreducible” or “inherent”).

Epistemic randomness: I’m happy to use this term in the way Chakraborty does, if I use it. I likely won’t.

Aleatory randomness/uncertainty: This is what I would likely use instead of epistemic randomness. Cases such as personalized drug interactions, arrow shaft quivering, rocky-boats billiards, earthquakes, currency-markets, the local weather precisely one year from now, and maybe even vigorous coin flips. (Note that this makes “Aleatoric Music” misnomer, etymology aside.)

Epistemic probability: The probability one assigns given what one knows, unless one is omniscient. In other words, it implies epistemic limitations, which is to say that it is to some degree subjective. But why shouldn’t this be a synonym for “subjective probability”?

Subjective probability: Hacking might have the right idea in avoiding this term. It can mean a lot of things. My most basic, go-to example is that, if I flip a coin right now, I know how it landed, but you still need to assign 50-50 to Heads-Tails as to how it landed. This may be better described as epistemic probability, as it involves a simple but perfectly valid probability model. Subjective probability might, as has been mentioned here already, involve a question like whether we live in a computer simulation, whether computers could ever be conscious, and so on. These questions may be answered following years of deep study and reflection, or a few seconds of thought (of the sort that is often encouraged of college students in introductory classes: “Now that I’ve given you a few minutes of exposure to the still-ongoing, decades-long controversy in this sub-sub-field, what do you think the correct answer is?”)

Subjective probability might refer to the aspect of Bayesian probability (which I won’t define here, as that is just what is treated in books like this one) that gets the most praise or criticism, depending on whom you talk to. Or it might simply mean any probability assignment that isn’t objective (in the sense of being the correct one, whatever that means; not to be confused with objective randomness); but this again may simply be identical to what people mean by “epistemic” probability when they say that all probability reduces to “epistemic” probability: i.e., all probability is subjective (see above Taleb survey). Or we might say if the subjective probability is the true probability, this is a matter of luck. Ugh. (Note that I replaced “objective” here with “true”; I think this is better, though still problematic, see below.)

At any rate, if I use this term, I will try to make it clear from context what I mean. But yeah, Hacking may be correct to avoid it.

See more on this term in the entry on “Objective probability.”

Objective probability: Avoid this term in favor of the “real” or “true” probability. Though these may be worth distinguishing. Namely, the true probability may not exist—may never exist—but rather the correct probability given a certain model may exist. (I’ve repeatedly discussed the difficulty, and it has been noted by others here, of determining by what standard to measure a probability; e.g., frequency vs. propensity). If we are using precisely the same model, then we are establishing a shared—and thus objective—model, which we may apply objectively in a technical sense, though we may fill it according to our own subjective/epistemic standpoint. For example, there is an objective way to define the prior probability in Bayes’ theorem, but what we put into that “prior” slot (if we put anything) may be subjective or epistemic (though, as this prior is transformed into a posterior probability that in turn become subsequent priors, those subsequent priors are determined by the theorem, particularly as iterations grow; this is the essential response, as I understand it, to the Bayesian skeptic).

Here is where I again differentiate “subjective” and “epistemic” probability: the former seems to allow for a vaguer, more opinion-oriented kind of subjectivity (as the term is colloquially used), while the latter carries an air of rigor. This may be misguided. What if “subjective probability” were replaced with “doxastic probability”, as doxa leans more in the direction of belief-as-opinion? Right? I suppose this would need to be aligned with doxastic logic (e.g., here’s an example I found via Google: “Dynamic Doxastic Probability Logic” by Irma Cornelisse, February 15, 2010).

Moral of the story: Try to be clear in context, and when that’s hard, aspire to clarity about the unclarity. To stay appropriately confused.

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Further Reading


  1. Download PDF here: Cited elsewhere as: S. Chakraborty, “On ‘Why’ and ‘What’ of Randomness,” DataCritica: International Journal of Critical Statistics, Vol. 3. No. 2, 2010, 1–13
  2. Chakraborty 1, Abstract
  3. Franklin cites these references: J. Keynes, Treatise on Probability (London, 1921), ch. 6; see L. Cohen, “Twelve questions about Keynes’ concept of weight,” British Journal for the Philosophy of Science 37 (1986): 263–78; R. O’Donnell, “Keynes’ weight of argument and Popper’s paradox of ideal evidence,” Philosophy of Science 59 (1992): 44–52.
  4. This turns out to align with certain views of God, namely from theodicy, i.e., to account for the problem of evil: If God knows what you’ll do before you do it, then how can be said to have free will? And if you don’t have free will, how can God send you to Hell for what you’ve done? Etc. One response to this is that God knows all there is to be known; the future has not yet happened yet, so future events are not among things that are known. The probabilistic element may result in problems here: How is it any better that God should create a system in which a human’s damnation comes down to a role of the dice?
  5. Atmanspacher H., Primas H. (2003) Epistemic and Ontic Quantum Realities. In: Castell L., Ischebeck O. (eds) Time, Quantum and Information. Springer, Berlin, Heidelberg
  6. There are three citations given here, which I haven’t read but look interesting: Armen Der Kiureghiana, Ove Ditlevsen, “Aleatory or epistemic? Does it matter?”, Structural Safety, Volume 31, Issue 2, March 2009, Pages 105–112.

    Hermann G. Matthies, “Quantifying Uncertainty: Modern Computational Representation of Probability and Applications, Extreme Man-Made and Natural Hazards in Dynamics of Structures, NATO Security through Science Series, 2007, 105-135, DOI: 10.1007/978-1-4020-5656-74

    Abhaya Indrayan, Medical Biostatistics (Second Edition), Chapman & Hall/CRC Press, 2008, pages 8, 673.

  7. For more on that, I recommend visiting the SEP entry Bayesian Epistemology.
  8. Referenced book: Understanding Post-Tonal Music by Miguel Roig-Francoli
  9. This link is worth reviewing as well.
  10. There really is a lot to talk about here. Even if for the sake of calling out myths like human observation—i.e., consciousness—creates reality. “Observation” is really just a technical term meaning something like “interaction with the world.” My favorite discussion of this topic is Sean Carroll’s Great Courses lecture series Mysteries of Modern Physics: Time, Lecture 8: “Time in Quantum Mechanics.” His lectures are refreshing both in their clarity and in their notable lack of exploiting fancy sounding concepts in order to “blow the mind” of us non-physicists. Carroll, who seems skeptical about the reality of irreducible randomness (or at least favors the many-worlds interpretation to the measurement problem; this does not involve hidden variables or epistemic restrictions, but it’s hard to say what this means for the notion of probability/randomness overall), manages to keep things both sober and interesting. For instance, one example he uses of an orange being “observed,” so that its wave function collapses, is a rock falling on it. (I’m noticing more and more popular communicators about physics going out of their way to use such examples, as a response to outlandish popular conceptions that hold consciousness itself as somehow responsible for reality.) A great supplement to the series is any conversation Carroll has had with David Albert; the first two are on YouTube, the third is Carroll’s Mindscape podcast: Science Saturday: Time’s Arrow || Science Saturday: Problems in Quantum Mechanics || Episode 36 (3/4/19): David Albert on Quantum Measurement and the Problems with Many-Worlds. The last one is my favorite, though all three are thoroughly enjoyable and thought-provoking.
  11. Ramsey, F. P., 1926, “Truth and Probability”, in Foundations of Mathematics and other Essays, R. B. Braithwaite (ed.), London: Kegan, Paul, Trench, Trubner, & Co., 1931, 156–198; reprinted in Studies in Subjective Probability, H. E. Kyburg, Jr. and H. E. Smokler (eds.), 2nd edition, New York: R. E. Krieger Publishing Company, 1980, 23–52; reprinted in Philosophical Papers, D. H. Mellor (ed.), Cambridge: Cambridge University Press, 1990.
  12. Philosophy of Science, Vol. 10, No. 4 (Oct., 1943), pp. 255–261, available at JSTOR.
  13. I happen to be writing another article parallel to this one, in which I illustrate that distinction as follows: Mary plans to marry the tallest man in town this Friday” could mean that Mary plans to wed a particular man—say, Pete—who, at this moment, happens to be the tallest man in town (de re); or that she plans to marry whoever happens to be the tallest man in town on Friday, which may or may not end up being Pete (de dicto). In the de re interpretation, “tallest man” is shorthand for referring to Pete, while under de dicto, “tallest man” refers to whoever happens to be the tallest man in town on Friday; should Pete leave town, for example, someone else will be the tallest man in town (even if that man also happens to be named Pete).
  14. Hacking is referring to a 1967 paper on Bernouilli by P.M. Boudot: “Probabilite et logic de l’argumentation selon Jacques Bernouilli,” Les études philosophiques. N.S. 28. 265–88.
  15. For more on this, see this Numberphile video featuring Persi Diaconis: How Random Is a Coin Toss?
  16. From Werner Heisenberg’s 1958 book Physics and Philosophy: The Revolution in Modern Science, p 53.
  17. His website is fun. And, for later viewing, here’s a debate on YouTube between Franklin and Norman Wildberger—Infinity: Does it Exist??
  18. Here, in addition to Keynes, Franklin cites E.T. Jaynes, whose book I’ll also dig into for this survey.
  19. I haven’t read McGrayne’s book, whose subtitle is: How Bayes’ Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy.
  20. Available in the 2011 essay collection Philosophy of Probability: Contemporary Readings, edited by Antony Eagle. It originally appeared as “Lecture 1” in von Mises’ volume, Probability, Statistic and Truth. I don’t own that book, but do own and have read much of the aforementioned essay collection, which is recommendable at the very least for the edutaining rebuttals against von Mises, and frequentism more broadly, from Richard Jeffrey and Alan Hájek!
  21. Maybe an understatement, given the degree of praise I’ve recently observed given to Tetlock’s work, e.g., in this discussion between Robert Wiblin and Tyler Cowen on the 80,000 Hours Podcast episode #45, 10/17/18. Tetlock has also been interviewed on that podcast, though I haven’t yet listened: #15, 11/20/17
  22. Popper, K. (1957), “The propensity interpretation of the calculus of probability, and the quantum theory,” in Observation and Interpretation, S. Körner, ed., Butterworth’s Scientific Publications, London, pp. 65–70. (It’s noted in the References section that: “Here, Popper, who had criticized quantum theory, summarizes his views to an audience of scientists concerned with foundations of quantum theory.”
  23. Cramér, H. (1946), Mathematical Methods of Statistics, Princeton University Press. (It’s noted in the References section that: “This marks the heyday of supreme confidence in confidence intervals over Bayesian methods, asserted as usual on purely ideological grounds, taking no note of the actual results that the two methods yield. Comments on it are in Appendix B and Jaynes (1976, 1986a).”
  24. Popper, K. (1959), ‘The propensity interpretation of probability’, Brit. J. Phil. Sci. 10, pp. 25–42. END-24) he defines the propensity interpretation more completely; now a propensity is held to be ‘objective’ and ‘physically real’ even when applied to the individual trial. In the following we see by mathematical demonstration some of the logical difficulties that result from a propensity interpretation. Popper complains that in quantum theory one oscillates between “… an objective purely statistical interpretation and a subjective interpretation in terms of our incomplete knowledge,” and thinks that the latter is reprehensible and the propensity interpretation avoids any need for it. He could not possibly be more mistaken. In Chapter 9 we answer this in detail at the conceptual level; obviously, incomplete knowledge is the only working material a scientist has! In Chapter 10 we consider the detailed physics of coin tossing, and see just how the method of tossing affects the results by direct physical influence.(Jaynes, p 61, Foonote #1)

    Exciting stuff! And good thoughts to keep churning in the background as we continue along past this survey. The term “propensity,” by the way, gets at a certain theoretical and philosophical framework for thinking about probability, defined in the aforementioned SEP entry, Interpretations of Probability, as:

    Like the frequency interpretations, propensity interpretations locate probability ‘in the world’ rather than in our heads or in logical abstractions. Probability is thought of as a physical propensity, or disposition, or tendency of a given type of physical situation to yield an outcome of a certain kind, or to yield a long run relative frequency of such an outcome. (Subsection 3.5)

    Jaynes touches on quantum theory here and there as he goes along, with a generally critical tone. I’ll jump over all that, landing at a section in Chapter 10 called “But What About Quantum Theory?,” where he says more about “physical” probability (among other things):

    Those who cling to a belief in the existence of “physical probabilities” may react to the above arguments by pointing to quantum theory, in which physical probabilities appear to express the most fundamental laws of physics. Therefore let us explain why this is another case of circular reasoning. We need to understand that present quantum theory uses entirely different standards of logic than does the rest of science.

    In biology or medicine, if we note that an effect E (for example, muscle contraction, phototropism, digestion of protein) does not occur unless a condition C (nerve impulse, light, pepsin) is present, it seems natural to infer that C is a necessary causative agent for E. Most of what is known in all fields of science has resulted from following up this kind of reasoning. But suppose that condition C does not always lead to effect E; what further inferences should a scientist draw? At this point, the reasoning formats of biology and quantum theory diverge sharply.

    In the biological sciences, one takes it for granted that in addition to C there must be some other causative factor F, not yet identified. One searches for it, tracking down the assumed cause by a process of elimination of possibilities that is sometimes extremely tedious. But persistence pays off; over and over again, medically important and intellectually impressive success has been achieved, the conjectured unknown causative factor being finally identified as a definite chemical compound. Most enzymes, vitamins, viruses, and other biologically active substances owe their discovery to this reasoning process.

    In quantum theory, one does not reason in this way. Consider, for example, the photoelectric effect (we shine light on a metal surface and find that electrons are ejected from it). The experimental fact is that the electrons do not appear unless light is present. So light must be a causative factor. But light does not always produce ejected electrons; even though the light from a unimode laser is present with absolutely steady amplitude, the electrons appear only at particular times that are not determined by any known parameters of the light. Why then do we not draw the obvious inference, that in addition to the light there must be a second causative factor, still unidentified, and the physicist’s job is to search for it?

    What is done in quantum theory today is just the opposite; when no cause is apparent one simply postulates that no cause exists—ergo, the laws of physics are indeterministic and can be expressed only in probability form. The central dogma is that the light determines not whether a photoelectron will appear, but only the probability that it will appear. The mathematical formalism of present quantum theory—incomplete in the same way that our present knowledge is incomplete—does not even provide the vocabulary in which one could ask a question about the real cause of an event.

    Biologists have a mechanistic picture of the world because, being trained to believe in causes, they continue to use the full power of their brains to search for them—and so they find them. Quantum physicists have only probability laws because for two generations we have been indoctrinated not to believe in causes—and so we have stopped looking for them. Indeed, any attempt to search for the causes of microphenomena is met with scorn and a charge of professional incompetence and “obsolete mechanistic materialism.” Therefore, to explain the indeterminacy in current quantum theory we need not suppose there is any indeterminacy in Nature; the mental attitude of quantum physicists is already sufficient to guarantee it. [Footnote excluded here.]

    This point also needs to be stressed, because most people who have not studied quantum theory on the full technical level are incredulous when told that it does not concern itself with causes; and, indeed, it does not even recognize the notion of ‘physical reality’. The currently taught interpretation of the mathematics is due to Niels Bohr, who directed the Institute for Theoretical Physics in Copenhagen; therefore it has come to be called ‘The Copenhagen interpretation’.

    As Bohr stressed repeatedly in his writings and lectures, present quantum theory can answer only questions of the form: ‘If this experiment is performed, what are the possible results and their probabilities?’ It cannot, as a matter of principle, answer any question of the form: “What is really happening when …?” Again, the mathematical formalism of present quantum theory, like Orwellian newspeak, does not even provide the vocabulary in which one could ask such a question. These points have been explained in some detail by Jaynes (1986d, 1989, 1990a, 1992a).

    We suggest, then, that those who try to justify the concept of ‘physical probability’ by pointing to quantum theory are entrapped in circular reasoning, not basically different from that noted above with coins and bridge hands. Probabilities in present quantum theory express the incompleteness of human knowledge just as truly as did those in classical statistical mechanics; only its origin is different.

    In classical statistical mechanics, probability distributions represented our ignorance of the true microscopic coordinates—ignorance that was avoidable in principle but unavoidable in practice, but which did not prevent us from predicting reproducible phenomena, just because those phenomena are independent of the microscopic details.

    In current quantum theory, probabilities express our own ignorance due to our failure to search for the real causes of physical phenomena; and, worse, our failure even to think seriously about the problem. This ignorance may be unavoidable in practice, but in our present state of knowledge we do not know whether it is unavoidable in principle; the ‘central dogma’ simply asserts this, and draws the conclusion that belief in causes, and searching for them, is philosophically naive. If everybody accepted this and abided by it, no further advances in understanding of physical law would ever be made; indeed, no such advance has been made since the 1927 Solvay Congress in which this mentality became solidified into physics. [Footnote excluded here.] But it seems to us that this attitude places a premium on stupidity; to lack the ingenuity to think of a rational physical explanation is to support the supernatural view.

    To many people, these ideas are almost impossible to comprehend because they are so radically different from what we have all been taught from childhood. Therefore, let us show how just the same situation could have happened in coin tossing, had classical physicists used the same standards of logic that are now used in quantum theory. (Jaynes 327–329)

    Here we have some particularly vicious criticism of the aforementioned Copenhagen interpretation. Also present is a criticism of a disdain for causes. I’m remind here of another book I likely would choose were I to add another to this survey: Judea Pearl’s excellent

    The Book of Why: The New Science of Cause and Effect (2018; cowritten with mathematician-turned-science-writer Dana Mackenzie), which I recently read and quite enjoyed. Pearl made massive contributions—I believe was the principal contributor—to the development of Bayesian networks (he even coined the term, in 1985), referred to in an above passage from Franklin. Pearl has since confessed, however, that, like so many others, he too had made the mistake of not always putting “causality first and probability second” (P&M 50), something he now believes led to Bayesian networks failing to “bridge the gap between artificial and human intelligence” (P&M 50); the missing ingredient—the one he should have insisted on rather than uncertainty—being causality. I won’t say any more about causality, but will note that it adds yet another wrinkle—fault line?—to the probabilistic landscape.

    Jaynes has yet more to say about quantum theory, but I think we get the idea.

    Unsurprisingly, he says more than I can possibly get into here about many of the other ideas we’ve been on the lookout for. I’ll dig around just a little more.


    Here’s my favorite representative tidbit on inductive probability, from a blurb in the References section, in response to an article by Popper & Miller:

    Popper, K. & Miller, D. W. (1983), “A proof of the impossibility of inductive probability,” Nature, 302, 687–688. They arrive at this conclusion by a process that we examined in Chapter 5; asserting an intuitive ad hoc principle not contained in probability theory. Written for scientists, this is like trying to prove the impossibility of heavier-than-air flight to an assembly of professional airline pilots. (Jaynes 689)


    Not all of the terms I’ve discussed here turn up in the book under those specific names (that I could find), not even in the Reference or Bibliography sections (e.g., “aleatory,” “ontic”), but, has been already noted, synonyms come up. And some terms just pop up a couple of times, like “personalistic” in these two excerpts:

    Likewise, we calculate the probability for obtaining various hands at poker; and we are so confident of the results that we are willing to risk money on bets which the calculations indicate are favorable to us. But underlying these calculations is the intuitive judgment that all distributions of cards are equally likely; and with a different judgment our calculations would give different results. Once again we are predicting definite, verifiable facts by “pure thought” arguments based ultimately on recognizing the “equally possible” cases; and yet present statistical doctrine, both orthodox and personalistic, denies that this is a valid basis for assigning probabilities! (Jaynes 394)

    Where I take “orthodox” to essentially mean “classical” or “frequentist” and “personalistic” to essentially mean “subjective” or “epistemic” or even “Bayesian.” Though such terms can of course be qualified according to who’s using them; note Jaynes’ use of quotes here in the sixth footnote of Chapter 10:

    Indeed, L. J. Savage (1962, p. 10225Savage, L. J. (1962), The Foundations of Statistical Inference, a Discussion, Methuen, London.

  25. Hanna, J., “Probabilistic Explanation and Probabilistic Causality,” PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, Vol. 1982, Volume Two: Symposia and Invited Papers (1982), pp. 181–193. Available at JSTOR.
  26. In contrast to the micro, or quantum, world; which, again, really just is the macro world—i.e., of clouds, rocks, ants, human brain, organisms, super organisms, social interactions, culture, ecology, and so on—we just need someone to reconcile the formal descriptions of these models, though it’s not clear, I don’t think, to what degree, if any, events in “macro” world are ontic, even if those of the micro world are ontic.
  27. Not to be confused with indeterminacy, which is, for example, when there is no fact of the matter about the location of a particle (and thus one cannot determine the location of the particle).
  28. Here Chakraborty refers us to: ref [1] S. Bolchan, “What is a random sequence?,” Ame. Math. Monthly, Jan, 2002; and ref [7] J. Ford, “How random is a coin toss?,” Physics Today, Vol. 36, 1983, p. 40–47

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