UNPRINCIPLED

Abstract

It is widely thought that chance should be understood in reductionist terms: claims about chance should be understood as claims that certain patterns of events are instantiated. There are many possible reductionist theories of chance, differing as to which possible pattern of events they take to be chance-making. It is also widely taken to be a norm of rationality that credence should defer to chance: special cases aside, rationality requires that one’s credence function, when conditionalized on the chance-making facts, should coincide with the objective chance function. It is a shortcoming of a theory of chance if it implies that this norm of rationality is unsatisfiable. The primary goal of this paper is to show, on the basis of considerations concerning computability and inductive learning, that this shortcoming is more common than one would have hoped.

1 Introduction

This paper is concerned with the rational relation between credence (degrees of belief) and chance (objective probability) and with reductionist theories of chance (according to which facts about chance supervene on non-chance-y facts). There is, of course, already a large and flourishing literature on these topics. Its dual centrepieces are: Lewis’s Best System account of chance, on which laws of chance are scientifically ideal summaries of patterns of events; and his Principal Principle, which says that the degrees of belief of rational agents must defer, in a certain sense, to facts about chance. ¹ In giving us a relation between chance and rational credence, the Principal Principle gives us interesting constraints on both relata: Lewis thought of the Principal Principle as telling us something about what it means to be rational and at the same time as encapsulating important truths about the concept of chance. ² Relative to a theory of chance, the Principal Principle picks out those prior credence functions that are rationally permitted. At the same time it gives us possible grounds for criticizing theories of chance. I suggest the following example: if we are thinking of theories of chance as something like summaries meant to be used by beings like us, there is something pathological about a theory of chance relative to which no priors count as rationally permitted.

Here I aim to make a contribution that is orthogonal to the directions pursued in most of the literature, making elementary use of some tools from the theory of computation and from the theory of inductive learning to clarify questions about which theories of chance are pathological in the sense just mentioned. It will emerge below that it is much more difficult than one might expect to find reductionist theories of chance that do not face difficulties of this sort.

Very roughly speaking: on Lewis’s Best System account, the chance laws at a world are just good scientific summaries of the patterns of events at that world; and his Principal Principle says that if you are rational, then to the extent that you are confident in a chance hypothesis it should guide your estimates of probability. So, taken together, the package seems to imply that in order to be rational, you need to be a certain sort of universal learner: whatever the chance laws at your world are, if you are a good scientist and you see enough data, you should become confident in something like those laws and so you should eventually mimic those laws in your estimates of probability. But in many contexts we know that there can be no universal learner. ³

Almost all of the discussion below will take place in the special setting in which possible worlds can be modelled by infinite binary sequences (simple enough to be tractable, complex enough to provide a picture of inquiry in which typical questions are never definitively settled by a finite number of observations). The framework we will be using is introduced in Sections 1.1–1.3.

Section 2 is concerned with reductionist theories of chance. For present purposes, such a theory of chance for a given space of possible worlds is a map (satisfying certain mild technical conditions) that assigns to (some) worlds of that space a probability measure on that space—for a world assigned such a probability measure, the measure plays the familiar role of chance laws (telling us the chance for various events to happen, unconditionally or conditionally on other events having happened). This section also introduces five very simple paradigm examples of reductionist theories of chance for our chosen set of worlds. These will be used throughout as illustrations.

Section 3 is concerned with a weak version of the Principal Principle, the Chance–Credence Principle (CCP). We will see that it is straightforward to characterize the priors that satisfy CCP with respect to any given theory of chance—and that some of our paradigm examples do not admit any priors that satisfy this principle.

There would be something unsatisfactory about an account of rationality that required rational agents to be able to perform some absolutely impossible task—squaring the circle, say. In the same way, if we are interested in the rationality of finite agents—e.g., human beings and the machines that they build—then there is something unsatisfactory about any account of rationality that requires agents to be able to perform tasks that are impossible for finite agents—solving the Halting Problem, say. ⁴ We will see in Section 4 that each of our paradigm examples of theories of chance is unsatisfactory in this way. So there is a challenge here for those attracted to reductionist theories of chance and to the Principal Principle: to develop and to defend examples of reductionist theories of chance that are CCP-satisfiable by computable priors (or to explain how norms of rationality can require agents to perform tasks that are impossible for computable agents).

Motivated in part by the picture sketched above of the relationship between the Lewisian package (Best System account of chance + the Principal Principle) and learning, Section 5 considers weaker analogs of CCP (and versions of them that require priors to be computable). The upshot is that if CCP is replaced by a weak enough principle along these lines, then some more reductionist theories of chance are consistent with Bayesian rationality. But many obstacles remain: for those interested in the combination of a reductionist theory of chance and a Principal Principle-style account of the relation between chance and credence there is interesting work to be done.

Appendix A consists of some remarks concerning the relation between the Principal Principle, some of its principal rivals, and CCP. Appendix B plays some Whac-A-Mole against the complaint that at various points, my discussion shows that I am being stiff-necked, closed-minded, naive, or unimaginative in my handling of conditional probabilities and null sets.

Ground-rules: all probability measures are real-valued and countably additive; the Axiom of Choice is taken for granted; and, for the most part, the notational conventions of [18] are followed. Non-reductionist theories of chance are neglected throughout—not because they are somehow immune from the problems encountered below, but because their treatment would require an extension of the present framework.

1.1 Main characters

1. a. The set of bits, $\mathsf {2}:=\{0,1 \}$ (note the sans serif font— $\mathsf {2}$ denotes a set, $2$ a number).

2. b. For each $n\in \mathbb {N}$ , $\mathsf {2}^n$ is the set of n-bit strings (we include the case of the empty string of zero bits).

3. c. The set of all binary strings: $\mathsf {2}^{<\omega } := \bigcup _{n=0}^\infty \mathsf {2}^n.$ If $\sigma $ and $\tau $ are binary strings we write $\sigma \tau $ for the string that results from concatenating $\sigma $ and $\tau $ (in that order). We call a binary string evenly split if it contains the same number of 0’s as it does 1’s.

4. d. Cantor space: the set $\mathsf {2}^\omega $ of all infinite binary sequences. ⁵ For $S \in \mathsf {2}^\omega ,$ we write $S(k)$ for the kth bit of S and $S{\upharpoonright }k$ for the string formed by concatenating the first k bits of $S.$ We call a binary sequence evenly split if the limiting relative frequency of 0’s in it is .5.

5. e. For any binary string $\tau ,$ we denote by $[\![\tau ]\!]$ the set of $S \in \mathsf {2}^\omega $ whose initial bits are given by $\tau .$ We call the $[\![\tau ]\!]$ basic subsets of $\mathsf {2}^\omega .$ ⁶ We use $\mathcal {B}$ to denote the $\sigma $ -algebra of subsets of $\mathsf {2}^\omega $ generated by the family of basic sets: $\mathcal {B}$ is the smallest family of subsets of $\mathsf {2}^\omega $ that includes all the basic sets and which is closed under taking complements, countable unions, and countable intersections. We call the members of $\mathcal {B}$ the Borel subsets of $\mathsf {2}^\omega .$ ⁷

6. f. The space $\mathcal {P}$ of Borel probability measures on $\mathsf {2}^\omega $ : each $\mu \in \mathcal {P}$ is a map from $\mathcal {B}$ to $[0,1]$ such that: (a) $\mu (\mathsf {2}^\omega )=1$ ; and (b) $\mu $ is countably additive (i.e., $\mu (\bigcup U_k)=\sum \mu (U_k),$ for any countable family $U_1$ , $U_2,$ … of pairwise disjoint Borel sets). We call a map m from the family of basic sets to $[0,1]$ a probability function if: (i) $m(\mathsf {2}^\omega )=1$ ; and (ii) for any binary string $\sigma $ , $m([\![ \sigma ]\!])=m([\![ \sigma 0]\!])+m([\![\sigma 1]\!]).$ ⁸ Any Borel probability measure determines via restriction a probability function. And the restriction in fact determines the probability measure: probability measures that agree on all basic sets agree on all Borel sets. ⁹ Further, every probability function can be extended to a Borel probability measure. ¹⁰

Example 1.1 (Delta-functions).

For any $S \in \mathsf {2}^\omega ,$ the delta-function concentrated on S is the Borel probability measure $\delta _S$ that for any binary string $\tau ,$ assigns the basic set $[\![ \tau ]\!]$ probability one if S begins with $\tau ,$ and otherwise assigns it probability zero.

Example 1.2 (Bernoulli measures).

For any $r \in (0,1),$ the Bernoulli measure with parameter r is the Borel probability measure $\nu _r$ such that for any binary string $\tau $ , $\nu _r([\![ \tau ]\!])=r^k(1-r)^\ell ,$ where k is the number of 0’s in $\tau $ and $\ell $ is the number of 1’s in $\tau .$ So according to $\nu _r,$ bits are selected by independent tosses of a coin with bias r in favour of 0’s. We call $\nu _{.5}$ the fair coin measure.

Example 1.3 (Hybrids).

At various points it will be helpful to have recourse to measures that initially behave like delta-functions before switching to the behaviour of the fair coin measure. For any binary string $\sigma $ we define the measure $\nu _{.5}^\sigma $ as follows: $\nu _{.5}^\sigma ([\![ \tau ]\!])$ is 1 if $\tau $ is an initial segment of $\sigma ,$ is $1/2^k$ if $\tau $ is a k-bit extension of $\sigma ,$ and is 0 otherwise.

Since there is a natural bijection between binary strings and the basic sets that they determine, when we are focusing on the action of probability measures on basic sets we will often simplify notation by writing $\mu (\tau )$ for $\mu ([\![\tau ]\!])$ etc. Similarly, if b is a bit, and $\tau $ is a binary string, the expression $\mu (b \, | \, \tau )$ stands for $\mu ([\![\tau b]\!] \, | \, [\![\tau ]\!]),$ the probability that $\mu $ gives for the next bit to be $b,$ conditional on the sequence beginning with the bits of $\tau .$

1.2 Computable numbers and measures

A binary sequence $S \in \mathsf {2}^\omega $ is computable if there is a Turing machine that on input of a natural number k gives as output the first k bits of $S.$

We say that a binary string $\sigma =b_1b_2\ldots b_n$ is a k-bit approximation to a number x in the unit interval if

$$ \begin{align*} \left| x - \sum_{j=1}^n \frac{b_j}{2^j} \right| \leq \frac{1}{2^k}. \end{align*} $$

A real number x in the unit interval is computable if there is a Turing machine that on input of a natural number k gives as output a k-bit approximation to $x.$

So, intuitively, a real number (or a sequence) is computable if and only if it can be computably approximated in a controlled way. Similarly, a real-valued function on a computably enumerable set is computable if there is a single machine that can approximate its value on any argument in a controlled way (this is stronger than just requiring that each of its values be computable). Specializing this to the case of probability measures—which, as we have seen, are determined by their restrictions to the basic subsets of $\mathsf {2}^\omega $ —it is natural to define a Borel probability measure $\mu $ on $\mathsf {2}^\omega $ to be computable if there is a Turing machine that on input of a binary string $\sigma $ and a natural number k gives as output a k-bit approximation to $\mu ([\![ \sigma ]\!]).$

Note that a delta-function measure $\delta _S$ is computable if and only if the sequence S that it is concentrated on is and that a Bernoulli measure $\nu _r$ is computable if and only if its parameter r is. Note, further, that since there are only countably many Turing machines, there can be only countably many computable sequences, computable real numbers, and computable probability measures.

1.3 Worlds and priors

We are going to use binary sequences to model certain simple possible worlds: time at these worlds is discrete, has a first instant, and is infinite towards the future; the state of one of these worlds at a time is encodable by a single bit. These worlds are too simple to be realistic, but allow us to study problems about credence and chance in a controlled setting (and of course they provide models of subsystems of more complex worlds).

We are going to work in a Bayesian framework in which agents satisfy three conditions.

1. Probabilism: Each agent begins life in a credal state encoded in a probability measure $\mu $ (the agent’s prior), defined over some salient set of possible worlds.

2. Evidential Regularity: Each agent’s prior must assign non-zero weight to any possible evidence E that the agent might acquire.

3. Conditionalization: The credal state of an agent with prior $\mu $ and total evidence E is encoded by $\mu (\cdot \, |\, E).$

In our setting, the natural explication of possible state of evidence is: a finite initial segment of a world. So we will call a probability measure $\mu \in \mathcal {P}$ a prior if it satisfies $\mu (\tau )>0$ for every binary string $\tau .$

2 Chance

Our focus here is on reductionist theories of chance. The spirit behind these (broadly Humean) approaches can be summarized as follows:

Philosophers are deeply divided on the status of modal truths in general and nomic truths in particular—truths related to physical possibility and necessity. Besides chance, prominent nomic phenomena include causation, counterfactuals, dispositions, and laws of nature. One approach to the nomic, going back to Hume and further to the medieval nominalists, holds that nomic facts about what could or would or must are always reducible to facts about what is. For the most part, Humeans do not deny the reality of nomic phenomena; they agree that there are laws of nature, chances, dispositions, etc. But they maintain that these things are derivative, determined by more fundamental, non-modal elements of reality. Thus Humeans might identify laws of nature with regularities in the history of physical events, and chances with relative frequencies. This ensures that whenever two possible worlds agree in non-nomic aspects, they also agree with respect to laws and chance. ¹¹

So what is a Humean reductionist theory of chance? The core idea is that: the chance facts are reducible to certain facts about patterns to be found in the whole panoply of events that happen in the history of our universe. What kinds of patterns? Broadly speaking, and unsurprisingly, they are stable and stochastic-looking or random-looking patterns. ¹²

In our setting, a complete history of a world is a binary sequence—a good candidate for an object that can be described in non-modal terms but which might or might not exhibit the sorts of patterns that could ground talk about chance. A probability measure $\nu $ on $\mathsf {2}^\omega $ can be thought of in the following terms: $\nu (0)$ gives a probability for the initial state of the world to be 0 and $\nu (1)$ gives the probability for it to be 1; and given any binary string $\tau $ , $\nu (0 \, | \, \tau )$ and $\nu (1 \, | \, \tau )$ give the probability for the next bit to be $0$ or $1,$ if $\tau $ gives the history so far. So we can picture $\nu $ as a book of instructions that tells Nature how to construct worlds: first, flip a coin of such and such bias to generate the first state; at any later time, flip a coin of this bias to generate the next state, if the history so far is that. More generally, for any (Borel) subset A of $\mathsf {2}^\omega ,$ and any binary string $\tau $ , $\nu (A\, |\, \tau )$ gives the probability for A to obtain, given that $\tau $ encodes an initial segment of the history of the world. ¹³

So in our setting, a reductionist theory of chance is an assignment of probability measures to (some) worlds. It is convenient to build a couple of weak technical conditions into the official definition.

Definition 2.1. A theory of chance is a (possibly merely partially defined) map $L : \mathsf {2}^\omega \to \mathcal {P}$ such that for any $\lambda $ in the image of L we have (i) that $L^{-1}(\lambda )$ is a Borel subset of $\mathsf {2}^\omega $ and (ii) that $\lambda ( L^{-1}(\lambda ))>0.$

Interpretation: if L is a theory of chance and $L(S) = \lambda $ for some $S \in \mathsf {2}^\omega $ and $\lambda \in \mathcal {P},$ then we think of L as saying that $\lambda $ encodes the facts about chance at $S.$ We refer to any $\lambda \in \mathcal {P}$ in the image of L as a law of chance of L (or a chance law of L). If $\lambda $ is a law of chance of $L,$ then we refer to the proposition (= set of worlds) $\Lambda := L^{-1}(\lambda )$ as the chance hypothesis determined by $\lambda .$ If, on the other hand, $L(S)$ is undefined, then L says that there are no facts about chance at $S.$ We will refer to the set of worlds $\Lambda _*$ at which L is undefined as the proposition that the world is lawless (it is the negation of the disjunction of all the chance hypotheses of the theory).

Clause (i) in Definition 2.1 requires that chance hypotheses be the sort of things that probabilities can be assigned to. ¹⁴ Clause (ii) rules out a certain sort of aberrant behaviour: a law of chance that says that there is zero chance that it is the law of chance. ¹⁵

In practice, we are most interested in theories of chance inspired by the heuristic considerations that underlie various classic philosophical programs for understanding chance reductively. Most prominent among these are: frequentist approaches, in which the chance of any event type is identified with the relative frequency with which that type of event occurs in some relevant reference class of events; and Best System approaches in which the facts about chance at a world are provided by the (in general stochastic) hypothesis that provides the best summary of facts at that world, according to ordinary scientific standards (very roughly speaking—the best hypothesis is the one that in some relevant sense ideally balances simplicity, logical strength, and fit with the data). ¹⁶ It will be helpful below to also have in mind a third (not especially popular) approach, super-determinism, on which the only laws of chance are deterministic (i.e., are encoded in delta-functions).

Consider the fair coin measure, $\nu _{.5},$ on $\mathsf {2}^\omega .$ What does it mean to think of it as the law of chance at a world $S \in \mathsf {2}^\omega $ ? Well, for starters, $\nu _{.5}$ tells you that the probability of $0$ occurring at any time is .5—and that this probability is independent of what happens at other times. In order to be acceptable to frequentists, a theory of chance must assign the fair coin measure to all and only worlds that are evenly split between 0’s and 1’s. Advocates of the Best System approach may recognize some exceptions to this rule: perhaps (in the setting of finite worlds) it sometimes makes sense to take the fair coin measure to give the law of chance at a world at which the relative frequency of 0’s is given by a number r that is close to .5, when r itself is sufficiently far from being simple. ¹⁷ And perhaps some evenly split sequences should be assigned a law of chance other than the fair coin measure—e.g., one might well want to assign a particularly simple $S\in \mathsf {2}^\omega $ the delta-function measure $\delta _S$ concentrated on S as its law of chance, whether or not S is evenly split. ¹⁸

It will be helpful to have in mind a few theories of chance that can be thought of as very simple-minded implementations of super-determinism, frequentism, and the Best System approach.

Example 2.2 (Super-Determinism).

Each $S \in \mathsf {2}^\omega $ is assigned as its chance law $\delta _S,$ the delta-function measure concentrated on $S.$

Example 2.3 (Computable Super-Determinism).

Each computable $S \in \mathsf {2}^\omega $ is assigned $\delta _S$ as its chance law. All other worlds are lawless.

Example 2.4 (Basic Frequentism).

If 0’s have relative frequency r in $S \in \mathsf {2}^\omega ,$ then the law of chance for S is the Bernoulli measure $\nu _r.$ If this relative frequency is not defined, S is lawless.

Example 2.5 (Computable Frequentism).

If 0’s have relative frequency r in $S \in \mathsf {2}^\omega $ and r is computable, then the law of chance for S is the Bernoulli measure $\nu _r.$ If the relative frequency of 0’s in S is uncomputable or undefined, then S is lawless.

There may seem to be little reason to opt for Computable Frequentism over Basic Frequentism: it doesn’t seem to follow from the thought that probabilities are relative frequencies that those frequencies must be given by computable numbers. But a restriction to computable chance laws is entirely natural on the Best System approach. If we are thinking of candidate chance laws for a world as something like putative scientific summaries of that world, with the actual chance law (if there is one) being the optimal such summary (as judged by ordinary scientific standards), then it would appear to be essential that candidate chance laws be finitarily specifiable. After all, what would it mean to compare the overall virtues of two infinite texts by the ordinary standards of scientific—or literary—criticism? To directly specify a binary sequence (or, equivalently, a real number in the unit interval) would be to specify the bits that make it up, one by one: an infinitary task. Turing launched modern theoretical computer science with the assertion that “The ‘computable’ numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means.” ¹⁹ Turing’s proposal is that the finitarily specifiable real numbers are the Turing-computable real numbers. A probability measure on $\mathsf {2}^\omega $ can be thought of as a certain sort of real-valued function on the countably infinite space of finite binary strings. Direct specification of such an object would be a doubly infinitary task, involving the specification of an infinite number of real numbers. But some such objects are finitarily specifiable: the Turing-computable measures (and the success of the Church–Turing thesis gives us reason to think that these are all of the finitarily specifiable such objects). So it is natural for those interested in Best System approaches to restrict their attention to theories in which all chance laws are computable. ²⁰

Example 2.6 (A Toy Best System Theory).

Any computable $S \in \mathsf {2}^\omega $ is assigned the delta-function measure concentrated on $S.$ Any incomputable S in which 0’s have computable relative frequency r is assigned the Bernoulli measure $\nu _r.$ All other S are lawless.

Remark 2.7. In a more plausible implementation of the best-system approach, it would be natural to further require that any sequence assigned a given computable probability measure as its law of chance must be algorithmically random relative to that measure. ²¹ It would also be natural to include further families of laws of chance: generalized Bernoulli measures (for which the bias of the coin used to determine the nth bit is allowed to depend on n); Markov chains (for which the bias of the coin used to determine the state at a time depends on the states at some fixed number of immediately preceding times); and so on.

Definition 2.8 (Proper and Improper Theories).

Let L be a theory of chance. We call a law of chance $\lambda $ of L with chance hypothesis $\Lambda $ proper if $\lambda (\Lambda )=1,$ otherwise we call $\lambda $ improper. We call L proper if all of its laws of chance are proper; otherwise we call L improper.

Remark 2.9. Each of the theories of chance mentioned above is proper. This follows from three facts. (a) Each delta-function $\delta _S$ assigns measure one to $\{S\}.$ (b) Each Bernoulli measure $\nu _r$ assigns measure one to the set of sequences in which 0’s have relative frequency r (the strong law of large numbers). (c) Each Bernoulli measure $\nu _r$ assigns measure zero to each countable set.

Example 2.10 (An Improper Theory of Chance).

As in Example 1.3, let $\nu _{.5}^0$ be the probability measure that is certain that the first bit will be 0 but thinks that all subsequent bits are sampled by tossing a fair coin. Consider the theory of chance that assigns the fair coin measure to all sequences that begin with 1 and assigns $\nu _{.5}^0$ all sequences that begin with 0. In this theory $\nu _{.5}^0$ is a proper chance law but the fair coin measure in an improper one.

3 Chance and credence

It has seemed to many that rationality requires there to be a certain sort of relation between chance and credence—that in certain situations, if you are rational, then your credence in an event must coincide with the chance of that event.

Knowing only that the chance of drawing a red ball from an urn is 0.95, everyone agrees, in accordance with the law of likelihood, that a guess of ‘red’ about some trial is much better supported than one of ‘not-red.’ But nearly everyone will go further, and agree that 0.95 is a good measure of the degree to which ‘red’ is supported by the limited data. ²²

$[$ T $]$ he chancemaking pattern in the arrangement of qualities must be something that would, if known, correspondingly constrain rational credence. Whatever makes it true that the chance of decay is 50% must also, if known, make it rational to believe to degree 50% that decay will occur. ²³

I will codify the constraint on priors suggested as follows.

Definition 3.1 (CCP).

Let L be a theory of chance. We say that a prior $\mu $ satisfies the Chance–Credence Principle (CCP) for L if: for any chance law $\lambda $ of L with chance hypothesis $\Lambda $ and for any Borel subset A of $\mathsf {2}^\omega $ we have

$$ \begin{align*} \mu(A \, | \, \Lambda ) = \lambda (A). \end{align*} $$

If there exists such a prior, we say that L is CCP-satisfiable; otherwise we say that it is CCP-unsatisfiable.

Remark 3.2 (Nomenclature).

The name Miller’s Principle is often given to principles in this neighbourhood. ²⁴ With more justice (and less snark) it might perhaps be called Hacking’s Principle, in light of Hacking’s somewhat earlier enunciation of a principle of this kind. ²⁵ But, really, it seems hopeless to try to identify the originator of an idea like this: as Hacking notes in introducing his version of the principle, it “seems to be so universally accepted that it is hardly ever stated.” ²⁶

CCP can be thought of as a weak form of the Principal Principle (see Appendix A). What does CCP require?

Definition 3.3. Let $\mu $ be a prior and let $\lambda $ be a law of chance of a theory of chance $L.$ We say that $\mu $ contains $\lambda $ as a nugget of truth if we can write $\mu = \nu + c\cdot \lambda $ with $c>0$ and $\nu $ a (possibly trivial) measure that considers the chance hypothesis $\Lambda $ of $\lambda $ a null set. ²⁷

Proposition 3.4. Let L be a theory of chance and let $\mu $ be a prior on $\mathsf {2}^\omega .$ Then $\mu $ satisfies CCP for L if and only if L has a countable number of chance laws, each of which is proper and contained in $\mu $ as a nugget of truth.

Proof. Suppose, first, that the chance laws of L are $\lambda _1$ , $\lambda _2,$ … (with chance hypotheses $\Lambda _1$ , $\Lambda _2,$ …), that each $\lambda _k$ is proper, and that $\mu = \nu + \sum c_k \cdot \lambda _k$ with each $c_k>0$ and $\nu $ a measure that considers each $\Lambda _k$ a null set. We show that $\mu $ satisfies CCP for $L.$ Fix $j=1,2,3,\ldots $ and a Borel set $A.$ We have

$$ \begin{align*} \mu(A \, | \, \Lambda_j ) & = \frac{\mu(A \, \& \, \Lambda_j)}{\mu(\Lambda_j)} \\[3pt] & = \frac{\nu(A \, \& \, \Lambda_j) + \sum c_k \cdot \lambda_k(A \, \& \, \Lambda_j)}{\nu( \Lambda_j) + \sum c_k \cdot \lambda_k( \Lambda_j)} \\[3pt] & = \frac{c_j \cdot \lambda_j (A \, \& \, \Lambda_j)}{c_j \cdot \lambda_j(\Lambda_j)} \\[3pt] & = \lambda_j(A), \end{align*} $$

where the first equality holds by definition; the second follows from the expansion for $\mu $ given above; the third follows from the fact that $\nu $ and the $\lambda _k$ other than $\lambda _j$ assign measure zero to $\Lambda _j$ ; and the fourth follows via the propriety of $\lambda _j.$

Suppose, on the other hand, that $\mu $ satisfies CCP for $L.$ Let $\lambda $ be one of the chance laws of L and let $\Lambda $ be the corresponding chance hypothesis.

1. (a) From the fact that CCP is satisfied, it follows in particular that the conditional probability $\mu (A \, | \, \Lambda )$ must be well-defined. So $\mu (\Lambda )>0.$ Since $\lambda $ was arbitrary, the corresponding fact must hold for each chance law of $L.$ Since the chance hypotheses corresponding to distinct chance laws must be disjoint, this means that L must have only countably many laws of chance.

2. (b) CCP tells us that a certain condition holds for all Borel sets A—so in particular, it holds when we set $A $ to $\Lambda .$ So L is proper:

   $$ \begin{align*} \lambda(\Lambda) & = \mu (\Lambda \, |\, \Lambda) \\ & = 1. \end{align*} $$

3. (c) Finally, let $\Lambda ^c$ be the complement of $\Lambda $ in $\mathsf {2}^\omega .$ Then $\mu |_\Lambda $ and $\mu |_{\Lambda ^c}$ are measures on $\mathsf {2}^\omega $ and $\mu = \mu |_\Lambda + \mu |_{\Lambda ^c}.$ Now, CCP says that for any measurable $A,$

   $$ \begin{align*} \lambda(A)& = \mu (A \, | \, \Lambda) \\ & = \frac{\mu(A \, \& \, \Lambda)}{\mu(\Lambda)} \\ & = \frac{1}{\mu(\Lambda)} \mu|_\Lambda (A). \end{align*} $$

   So $\mu = \mu |_{\Lambda ^c} + c \cdot \lambda ,$ where $c:=\mu (\Lambda )>0$ : $\mu $ contains $\lambda $ as a nugget of truth.

Corollary 3.5. Let L be a proper theory of chance with countably many chance laws $\lambda _1$ , $\lambda _2,$ … (with chance hypotheses $\Lambda _1$ , $\Lambda _2,$ …). A prior $\mu $ satisfies CCP for L if and only if it can be written in the form

$$ \begin{align*} \mu = \nu + \sum c_k\cdot \lambda_k, \end{align*} $$

with each $c_k>0$ and $\nu $ a (possibly trivial) measure on $\mathsf {2}^\omega $ such that $\nu (\Lambda _k)=0$ for each $k.$

Corollary 3.6. A theory of chance is CCP-satisfiable if and only if it is proper and has only countably many laws of chance.

Proof. The left to right implication follows immediately from the preceding proposition. For the other half, let L be a proper theory of chance with chance laws $\lambda _1$ , $\lambda _2,$ …. It suffices to show the existence of a prior $\mu $ that contains each of the $\lambda _k$ as a nugget of truth.

Let $\mu _0 = \sum \frac {1}{2^k} \lambda _k.$ Let $M := \{ \tau \in \mathsf {2}^{<\omega } \, | \, \mu _0([\![\tau ]\!])=0\}.$ If M is empty, then $\mu _0$ is a prior and we can take $\mu =\mu _0.$ Otherwise, enumerate M as $\tau _1$ , $\tau _2,$ … (M is countable, being a subset of $\mathsf {2}^{<\omega },$ and is infinite, since any extension of a string in M is in M). Then, recalling the notation of Example 1.3, $\mu _1 := \sum \frac {1}{2^k} \nu ^{\tau _k}_{.5}$ is a probability measure concentrated on the L-lawless sequences. So we can take $\mu = \frac {1}{2} \mu _0 + \frac {1}{2} \mu _1.$

Of our examples of theories of chance from Section 2, Super-Determinism and Basic Frequentism are not CCP-satisfiable (they have too many chance laws), but Computable Super-Determinism, Computable Frequentism, and the Toy Best System account are. ²⁸

4 Chance, credence, and computability

As noted above, fans of the Principal Principle like to present it as Janus-faced, both giving us substantive information about the nature of chance and giving us a substantive constraint on rational Bayesian priors. From this perspective, something has gone wrong if we find no computable prior satisfies CCP for a certain reductionist theory of chance: the theory of chance should be discarded; we need to replace CCP with some weaker condition relating credence and chance; or we need to provide an account of the normative force of a putative requirement of rationality that requires agents to perform supra-computable tasks.

Definition 4.1 (CCCP).

Let L be a theory of chance and let $\mu $ be a prior. We say that $\mu $ satisfies the Computable Chance–Credence Principle (CCCP) for L if $\mu $ is computable and satisfies CCP for $L.$

Definition 4.2. We call a theory of chance CCCP-satisfiable if there is a prior satisfying CCCP for it; otherwise we call it CCCP-unsatisfiable.

We are going to see that there are nontrivial obstructions to CCCP-satisfiability. Indeed, none of our paradigm examples of theories of chance are CCCP-satisfiable.

4.1 Learning and delta-functions

Our first objective will be to show that Computable Super-Determinism and the Toy Best System Theory are not CCCP-satisfiable. We will approach this result indirectly via one of the classic computer science models of inductive learning (this will give us purchase in Section 4.3 on the question of what sort of theories are CCCP-satisfiable). ²⁹

Definition 4.3. An extrapolating machine is a computable map $m: \mathsf {2}^{<\omega }\to \{0,1\}.$

Definition 4.4. Let m be an extrapolating machine and let $S\in \mathsf {2}^\omega .$ We say that m NV-learns S if there is an N such that $m(S{\upharpoonright }n)=S(n+1)$ for all $n>N.$

Informal picture: a learning agent is being fed an infinite binary data stream S bit-by-bit and each time a bit is revealed, the agent makes a guess as to the identity of the next bit. An extrapolating machine m is a computable strategy that an agent might use for arriving at such guesses: when given a binary string $\tau $ (a finite data set), m outputs a bit $m(\tau ).$ Extrapolating machine m successfully NV-learns S just in case from some point onwards in processing $S,$ all of m’s guesses are correct (here NV=next value). Note that any binary sequence NV-learned by an extrapolating machine must be computable.

Proposition 4.5 (Putnam [54]a,b).

No extrapolating machine NV-learns each computable sequence.

Proof. Let $m_0$ be an extrapolating machine and define the binary sequence S as follows: the first bit of S is $1- m_0(\varnothing )$ (the opposite of what $m_0$ predicts on empty input); and in general, $S(k+1)$ is the opposite of what $m_0$ predicts when shown the first k bits of S. S is computable (since $m_0$ is). And $m_0$ does not NV-learn S (since it never predicts a bit correctly).

Given a computable prior $\mu $ we can define an extrapolating machine $m_\mu $ as follows. Let $\kappa $ be a computable number in the open unit interval that is not in the set $X_\mu :=\{ \mu (b\, | \, \sigma ) \, | \, b\in \mathsf {2}, \sigma \in \mathsf {2}^{<\omega }\}$ (i.e., $\mu $ never uses $\kappa $ as the conditional probability that the next bit will be 0 or that it will be 1). ³⁰ Define $m_\mu : \mathsf {2}^{<\omega } \to \mathsf {2}$ as follows: on input of binary string $\sigma $ , $m_\mu $ outputs 0 if $\mu (0 \, | \, \sigma )> \kappa $ and outputs 1 if $\mu (0 \, | \, \sigma )< \kappa .$ Since $\kappa $ and $\mu $ are computable, so is $m_\mu .$ ³¹

Proposition 4.6. Let $\mu $ be a prior on $\mathsf {2}^\omega $ and let S be a binary sequence. If $\mu = \nu + c\cdot \delta _S,$ where $c>0$ and $\nu $ is a measure, then $m_\mu $ NV-learns $S.$

Proof. We can assume without loss of generality that $\nu (\{S\})=0.$ We then have

$$ \begin{align*} \mu (S(n+1) \, | \, S{\upharpoonright}n) & = \frac{\mu( S{\upharpoonright}(n+1))}{\mu(S{\upharpoonright}n)} \\ & = \frac{\nu(S{\upharpoonright}(n+1)) + c \cdot \delta_S (S{\upharpoonright}(n+1))}{\nu(S{\upharpoonright}n) + c \cdot \delta_S (S{\upharpoonright}n)} \\ & = \frac{\nu(S{\upharpoonright}(n+1)) + c}{\nu(S{\upharpoonright}n) + c}, \end{align*} $$

since $\delta _S(S{\upharpoonright }n)=1$ for all $n.$ Since $\nu (\{S\})=0,$ by making n sufficiently large we can make $\nu (S{\upharpoonright }n)$ —and therefore also $\nu (S{\upharpoonright }(n+1))$ —as small as we like. ³² So by making n sufficiently large, we can make $\mu (S(n+1)\, | \, S{\upharpoonright }n)$ as close to one as we like. So for sufficiently large n, $m_\mu $ always predicts the next bit of S correctly (i.e., $m_\mu $ NV-learns S—no matter what choice of $\kappa $ we made in defining $m_\mu $ ).

Proposition 4.7. Computable Super-Determinism and the Toy Best System Theory are CCCP-unsatisfiable.

Proof. Any prior $\mu $ on $\mathsf {2}^\omega $ that satisfied CCCP for one of these theories would in particular satisfy CCP for that theory—and so, by Proposition 3.4, would contain each computable delta-function measure as a nugget of truth. But then by Proposition 4.6, the extrapolating machine $m_\mu $ induced by $\mu $ would NV-learn each computable sequence. But Proposition 4.5 shows that to be impossible.

So Computable Super-Determinism and the Toy Best System Theory, although CCP-satisfiable, are CCCP-unsatisfiable: they admit priors that satisfy the weak form of the Principal Principle that we have been considering; but all such priors are uncomputable.

Remark 4.8. Adleman and Blum [1] show that if A is an oracle, then there exists a Turing machine with access to A that NV-learns every computable sequence if and only if A is high (i.e., A enables the computation of a function that grows more quickly than any computable function). So the problem of NV-learning all computable sequences is unsolvable, but strictly easier than the Halting Problem.

4.2 Learning and Bernoulli measures

The above discussion shows that Computable Super-Determinists must rein in their ambitions, maintaining that only a well-behaved proper subset of computable delta-function measures can correspond to chance laws, if they want their theory to support the existence of a prior satisfying CCCP. We will see next that Computable Frequentists must likewise fall back to a position on which only some of the computable Bernoulli measures arise as laws of chance, if they want their theory to support the existence of a prior satisfying CCCP. And, of course, fans of the Toy Best System will be driven to make both moves at once. ³³

We begin by recalling some basic notions from the theory of algorithmic randomness deriving from [48]. A family $\mathcal {U}=\{U_k \}_{k\in \mathbb {N}}$ of subsets of $\mathsf {2}^\omega $ is called a uniformly effective family of open sets if there is a Turing machine that on input $k \in \mathbb {N}$ outputs a sequence $\tau ^k_1$ , $\tau ^k_2,$ … of binary strings with the feature that

$$ \begin{align*} U_k = \bigcup_{j=1}^\infty \,\, [\![\tau^k_j]\!]. \end{align*} $$

That is: each $U_k$ can be effectively approximated as a union of basic subsets of $\mathsf {2}^\omega $ ; and this approximation is uniform in $k,$ in the sense that a single machine can handle each of the $U_k.$ ³⁴ Let $\nu $ be a computable probability measure on $\mathsf {2}^\omega .$ Then an effective $\nu $ -null set is a subset T of $\mathsf {2}^\omega $ that can be written in the form $T=\bigcap U_k,$ for some uniformly effective family of open sets, $\{U_k\},$ satisfying $\nu (U_k)\leq 2^{-k}.$ ³⁵ We say that a binary sequence is $\nu $ -Martin-Löf random if it does not belong to any effective $\nu $ -null set and we denote by $\mathsf {ML}_\nu $ the set of such sequences. Note that each computable $\nu \in \mathcal {P}$ assigns measure one to its $\mathsf {ML}_\nu .$ ³⁶

Proposition 4.9. Let $R=\{ r_k\}_{k\in I}$ be a set of computable binary sequences and for each $k\in I$ let $\nu _k$ be the Bernoulli measure whose parameter admits $r_k$ as a binary expansion. If $\mu $ is a prior that can be written in the form $\mu = \nu + \sum c_k \cdot \nu _k$ (with $\nu $ a possibly trivial measure and each $c_k>0$ ), then there is an extrapolating machine that NV-learns each $r_k.$

Proof. (1) We claim that for any $m \in I$ and any binary sequence $S,$ if $S\in \mathsf {ML}_{\nu _{m}}$ , then $S\in \mathsf {ML}_{\mu }.$ For suppose that $S \notin \mathsf {ML}_{\mu }.$ Then there exists a uniformly effective family of open sets, $U_1$ , $U_2,$ …, with $\mu (U_k)\leq 2^{-k}$ (for each $k\in \mathbb {N}$ ) and $S \in \bigcap _{k=1}^\infty U_k.$ Choosing j large enough to ensure that $1/c_m \leq 2^{j},$ we have

$$ \begin{align*} \nu_{m} (U_k) & \leq \frac{1}{c_m} \mu (U_k) \\ & \leq 2^{-k + j} \end{align*} $$

for each $k\in \mathbb {N}.$ So if we take $V_k:= U_{k+j},$ then the $V_k$ form a uniformly effective family of open sets and satisfy $\nu _{m}(V_k) \leq 2^{-k},$ with $S \in \bigcap V_k=\bigcap U_j.$ So $S \notin \mathsf {ML}_{\nu _{m}}.$

(2) Next, we invoke a result due to Porter: for any computable measure $\lambda $ on $\mathsf {2}^\omega ,$ there is a computable measure $\zeta $ on $\mathsf {2}^\omega $ such that for each $r \in [0,1],$ if $\mathsf {ML}_\lambda \cap \mathsf {ML}_{\nu _r} \neq \varnothing ,$ then $r \in \mathsf {ML}_\zeta .$ ³⁷ Applied to our case, this implies that there exists a computable probability measure $\zeta $ such that $R \subseteq \mathsf {ML}_{\zeta }.$

(3) It follows that $\zeta $ must assign positive probability to each $\{ r_k\}$ : otherwise, since $\zeta $ and $r_k$ are both computable, we could choose initial segments $\tau _1$ , $\tau _2,$ … of $r_k$ so that the family of basic sets $[\![\tau _1]\!]$ , $[\![\tau _2]\!],$ … would be uniformly effectively open and satisfy $\zeta ([\![\tau _m]\!])< 2^{-m},$ with $r_k \in \bigcap [\![\tau _j]\!],$ contradicting $r_k \in \mathsf {ML}_{\zeta }.$ It then follows from Proposition 4.6 that the extrapolating machine $m_\zeta $ determined by $\zeta $ must NV-learn each $r_k.$

Corollary 4.10. Computable Frequentism and the Toy Best System are CCCP-unsatisfiable.

Remark 4.11. Vitányi and Chater [63] develop a model of learning in which an agent viewing a binary data stream attempts to guess a code number for a Turing machine that calculates a probability measure on $\mathsf {2}^\omega $ relative to which the data stream is Martin-Löf random. Barmpalias et al. [5, theorem 1.6] show that the set of computable Bernoulli measures is not learnable in this sense—indeed, they further show that this set is learnable relative to an oracle if and only if that oracle is high. It is natural to conjecture that an analogous result holds in our setting: there exists a prior satisfying CCP for computable Frequentism that is computable relative to a given oracle if and only if that oracle is high.

4.3 Satisfying CCCP

So in order for a prior to satisfy CCCP for a given theory of chance, the set sequences on which the theory’s delta-function laws of chance are concentrated and of parameters of its Bernoulli measure laws of chance must both be NV-learnable. Here we consider the contours of NV-learnability. We denote by $NV(m)$ that set of sequences NV-learned by extrapolating machine $m.$

Definition 4.12. A set $\mathcal {F}$ of binary sequences is dense if for each binary string $\tau ,$ there is a sequence S in $\mathcal {F}$ with $\tau $ as an initial segment. ³⁸

Definition 4.13. A set $\mathcal {F}$ of binary sequences is uniformly computable if there is an enumeration $S_1$ , $S_2$ , $S_3,$ … of the elements of $\mathcal {F}$ such that the map that sends $(k,\ell )\in \mathbb {N}^2$ to $S_k(\ell )$ is computable.

Remark 4.14. It is immediate that each member of a uniformly computable set of sequences is computable. A straightforward diagonalization argument shows that the set of computable sequences is not itself uniformly computable. And since any finite subset of $\mathsf {2}^\omega $ is uniformly computable and the union of any two uniformly computable sets of sequences in uniformly computable, it follows that any uniformly computable family of binary sequences must exclude infinitely many computable binary sequences. Moral: to say that a set of sequences is uniformly computable is to say that it is computationally tractable in a certain sense.

Proposition 4.15. Let $\mathcal {F}$ be a set of binary sequences. The following are equivalent:

1. i) $\mathcal {F}$ is uniformly computable and dense.

2. ii) There is an extrapolating machine m such that $\mathcal {F}=NV(m).$

Proof. Let $\mathcal {F}\subset \mathsf {2}^\omega $ be uniformly computable and dense. Fix an enumeration $S_1$ , $S_2,$ … of the members of $\mathcal {F}$ such that the map $(k,\ell ) \mapsto S_k(\ell )$ is computable. Define an extrapolating machine m as follows: on input of an n-bit binary string $\tau $ , m finds the least k such that $S_k{\upharpoonright }n=\tau $ and gives output $m(\tau )=S_k(n+1)$ (such a k always exists since $\mathcal {F}$ is dense). For any $S_k\in \mathcal {F},$ we can find n large enough so that there is no $j <k$ with $S_j \neq S_k$ and $S_j{\upharpoonright }n=S_k{\upharpoonright }n.$ It follows that m NV-learns each sequence in $\mathcal {F}.$ Suppose, on the other hand, that m NV-learns S and let $\tau $ be an initial segment of S that contains all of the bits that m predicts incorrectly. On input $\tau $ , m uses some $S_k$ that has $\tau $ as an initial segment to predict that next bit. This prediction is correct, so m also uses $S_k$ to predict the next bit—and all subsequent bits. So $S=S_k.$ So each sequence NV-learned by m is in $\mathcal {F}.$ So $\mathcal {F} = NV(m).$

Suppose that there is an extrapolating machine m such that $\mathcal {F}=NV(m).$ Then $\mathcal {F}$ must be dense, since it must include, for each binary string $\tau ,$ the sequence $S_m^\tau $ that has $\tau $ as an initial segment and in which all subsequent bits are chosen to vindicate m’s predictions. Further, if $\tau _1$ , $\tau _2$ , $\tau _3,$ … is a computable enumeration of the binary strings, then $S_m^{\tau _1}$ , $S_m^{\tau _2}$ , $S_m^{\tau _3},$ … is an enumeration of the sequences NV-learned by m and the map $(k, \ell ) \mapsto S_m^{\tau _k}(\ell )$ is computable, since m and our enumeration of the strings are both computable. So $\mathcal {F}$ is uniformly computable.

Corollary 4.16. Let $\mathcal {F}$ be a set of binary sequences. There is an extrapolating machine that NV-learns each sequence in $\mathcal {F}$ if and only if $\mathcal {F}$ is a subset of a uniformly computable set of binary sequences.

Proof. Note that the union of a uniformly computable set of sequences with the set of periodic sequences (i.e., those sequences that can be written in the form $\sigma \sigma \sigma \ldots $ for some binary string $\sigma $ ) is both uniformly computable and dense.

Remark 4.17. This corollary gives a version of one of the two standard characterizations of NV-learnability. ³⁹ The other is: A set $\mathcal {F}$ of computable binary sequences is a subset of some $NV(m)$ if and only if it is a subclass of an abstract complexity class. ⁴⁰ One gloss that has been given to this latter characterization: “It says, in essence, that the [computably] extrapolable sequences are the ones that can be computed rapidly.” ⁴¹ Perhaps a more accurate gloss would be: for any computably NV-learnable set S of sequences, there is an at least somewhat natural notion of rapidly computable relative to which every sequence in S is rapidly computable.

All of this may seem congenial to fans of the Best System approach. True, they may have to recognize fewer chance laws than they might have originally hoped—but to the extent that we can think of there being some sort of complexity cut-off that rules out certain delta-function measures and Bernoulli measures as respectable chance laws, that may seem to uphold the spirit of the Best System approach. (But here it is important to keep in mind the arbitrariness entailed by the considerations canvassed in Remark 4.14.)

Let us end this discussion on a positive note.

Definition 4.18. We say that a set $\mathcal {M}$ of probability measures on $\mathsf {2}^\omega $ is uniformly computable if there is an enumeration $\nu _1$ , $\nu _2,$ … of the elements of $\mathcal {M}$ and a Turing machine that on input of a binary string $\sigma $ and non-zero natural numbers j and $k,$ give as output a k-bit approximation (as in Section 1.2) to $\nu _j (\sigma ).$

Proposition 4.19. If L is a proper theory of chance whose set of chance laws is uniformly computable and includes a prior, then L is CCCP-satisfiable.

Proof. Let $\lambda _1$ , $\lambda _2,$ … be an enumeration of the chance laws of L in virtue of which these laws are uniformly computable. Consider the measure

$$ \begin{align*} \mu := \sum \frac{1}{2^k} \lambda_k. \end{align*} $$

It is a prior (since by assumption one of the $\lambda _k$ ’s is). It is immediate that $\mu $ satisfies CCP for $L.$ And it is straightforward to show that $\mu $ is computable. ⁴²

5 Will we never learn?

A reaction that people sometimes have to the Principal Principle: Who cares how my prior behaves conditional on an event E like that of the coin being fair? If I am a true Humean, I should regard this E as something that could never be part of my evidence, since it is a holistic fact about the history of the entire universe. ⁴³

Lewis has an answer to this challenge. ⁴⁴

To the believer in chance, chance is a proper subject to have beliefs about. Propositions about chance will enjoy various degrees of belief, and other propositions will be believed to various degrees conditionally upon them. ⁴⁵

To the subjectivist who believes in objective chance, particular or general propositions about chance are nothing special. We believe them to varying degrees. As new evidence arrives, our credence in them should wax and wane in accordance with Bayesian confirmation theory. It is reasonable to believe such a proposition, like any other, to the degree given by a reasonable initial credence function conditionalized on one’s present total evidence. ⁴⁶

We are interested in the Principal Principle, not because we want to be prepared just in case we should learn the truth of some law of chance, but because chance laws can be confirmed and disconfirmed: we have varying degrees of credence in various chance laws and the Principal Principle is a constraint on the rational connection between those theoretical beliefs and concrete expectations about future events. ⁴⁷

Learnability is in fact an important (if largely subterranean) Lewisian theme in the paper in which the Principal Principle made its debut. ⁴⁸ He has recourse to non-standard-valued priors because he thinks that rational priors must be regular (i.e., assign non-zero probability to each non-trivial proposition):

it is required as a condition of reasonableness: one who started out with an irregular credence function (and who then learned from experience by conditionalizing) would stubbornly refuse to believe some propositions no matter what the evidence in their favor. ⁴⁹

In a post-script he suggests that “if we start with a reasonable initial credence function and do enough feasible investigation, we may expect our credences to converge to the chances.” ⁵⁰

Now, within the standard framework that we have been working in, only Bayesian agents who assign non-zero prior probability to a chance hypothesis can become more confident of that chance hypothesis through experience. And within our standard framework, no Bayesian agent can assign non-zero prior probability to each chance hypothesis of a theory of chance that has uncountably many laws of chance. It is natural to think that something has gone wrong here. Surely Basic Frequentism, despite having uncountably many laws of chance, should provide an extraordinarily hospitable environment for a Bayesian agent attempting to learn the chance facts!

One option would be too simply excuse priors from having to defer to chance laws when they assign the corresponding chance hypothesis probability zero. ⁵¹

Definition 5.1 (CCP $^*$ ).

Let L be a theory of chance and let $\mu $ be a prior. We say that $\mu $ satisfies CCP $^*$ for L if: for any Borel subset A of $\mathsf {2}^\omega $ and for any chance law $\lambda $ such that $\mu $ assigns positive probability to $\Lambda := L^{-1}(\lambda ),$ we have $\mu (A \, | \, \Lambda ) = \lambda (A).$

Relative to Basic Frequentism, the fair coin measure satisfies CCP $^*$ (but not CCP of course). Someone who begins life with this prior is certain from birth that the chance of a 0 at any particular future time is 1/2 and remains certain of this no matter how strongly the data speak against this hypothesis and in favour of others. ⁵² To accept CCP $^*$ as an explication of the idea that a rational agent’s credence function should defer to chance is to give up entirely on the idea that part of what deference to chance involves is being open to learning chance hypotheses. ⁵³

I think that we should be interested in finding variants of CCP that require priors to be open to the learning of chance hypotheses. ⁵⁴ To this end, let us zoom out. The thought behind the Best System approach is that a chance law of a world is a sort of optimal scientific summary of the pattern of events at that world (relative to the ordinary standards of scientific practice). The thought behind the Principal Principle is that (in ordinary circumstances at least) to the extent that you are confident that the chance law of your world is given by $\lambda ,$ your credences should be close to the $\lambda $ -chances. Suppose that a given prior $\mu $ satisfies at least the spirit of the Principal Principle and embodies the canons of scientific rationality (preferring simpler theories to more complex ones, etc.). ⁵⁵ Then it seems that if $\mu $ sees larger and larger portions of a world with chance law $\lambda _0,$ it should become more and more confident that the law of chance is $\lambda _0$ -like (since $\mu $ is a good scientist, it will eventually home in on the best scientific description of its world) and hence, upon being conditionalized on larger and larger data sets, the probabilities that $\mu $ assigns events will approach those that $\lambda _0$ assigns them (since $\mu $ satisfies the spirit behind the Principal Principle).

Let L be a proper theory of chance and let $\lambda $ be a chance law of L with corresponding chance hypothesis $\Lambda =L^{-1}(\lambda ).$ Let S be typical in $\Lambda .$ Our line of thought above tells us that as $n\to \infty ,$ we expect that $\mu (\cdot \, |\, S{\upharpoonright }n)$ comes closer and closer to $\lambda (\cdot \, | \, S{\upharpoonright }n)$ in some sense. So let us consider generalizations of CCP of the form: there exists a prior $\mu $ such that for every chance law $\lambda ,$ for $\lambda $ -almost all sequences S in the corresponding chance hypothesis, the $\mu (\cdot , |\, S{\upharpoonright }n)$ converge to the $\lambda (\cdot \, |\, S{\upharpoonright }n)$ as $n \to \infty .$ The question is: What notion of convergence should we demand here?

5.1 Merging

Recall that the total variation distance between probability measures $\lambda $ and $\nu $ on $\mathsf {2}^\omega $ is $\sup _{A \in \mathcal {B}} | \nu (A) - \lambda (A) |.$ ⁵⁶ The following is a specialization to our context of the notion of merging introduced in [14] and subsequently widely discussed by statisticians and game theorists.

Definition 5.2 (Merging).

Let $\lambda $ and $\mu $ be probability measures on $\mathsf {2}^\omega .$ We say that $\mu $ merges with $\lambda $ if there is a set of sequences of $\lambda $ -measure one such that for each S in that set:

$$ \begin{align*} \lim_{n \to \infty} \sup_{A \in \mathcal{B}} |\mu(A \, | \, S{\upharpoonright}n)- \lambda(A \, | \, S{\upharpoonright}n) | = 0. \end{align*} $$

Definition 5.3 (M-CCP).

Let L be a theory of chance. We say that a prior $\mu $ on $\mathsf {2}^\omega $ satisfies the Merging Chance–Credence Principle (M-CCP) if $\mu $ merges with each chance law of $L.$

If a prior $\mu $ merges with a proper law of chance $\lambda $ of a theory of chance, then we can take the set of sequences that witnesses merging to be a subset of the chance hypothesis of $\lambda .$

Definition 5.4. We call a theory of chance L M-CCP-satisfiable if there is a prior that satisfies M-CCP for $L.$ If there is a computable prior with this feature, we call L M-CCCP-satisfiable.

A pair of classic results clarify what it takes for a prior to merge with a probability measure $\nu .$ Recall that if $\nu _1$ and $\nu _2$ are measures on a measurable space, then we say that $\nu _1$ is absolutely continuous with respect to $\nu _2$ if for each measurable set A, $\nu _1(A)>0$ implies $\nu _2(A)>0.$ In this case we write

Remark 5.5. Note that if we have some measures satisfying $\mu = \nu _1 + \nu _2,$ then So, in particular, if a prior $\mu $ contains a proper law of chance $\lambda $ as a nugget of truth, then

Proposition 5.6 (Blackwell and Dubins [14]).

Let $\lambda $ be a probability measure on $\mathsf {2}^\omega $ and let $\mu $ be a prior on $\mathsf {2}^\omega .$ If then $\mu $ merges with $\lambda .$

Proposition 5.7 (Lehrer and Smorodinsky [44]).

Let $\lambda $ be a probability measure on $\mathsf {2}^\omega $ and let $\mu $ be a prior on $\mathsf {2}^\omega .$ If $\mu $ merges with $\lambda ,$ then

Each prior merges with uncountably many probability measures. ⁵⁷ But only countably many of these can be chance laws of any given reductionist theory of chance.

Proposition 5.8. Let L be a theory of chance. If L is M-CCP satisfiable, then L has only countably many chance laws.

Proof. Suppose that $\mu $ satisfies the M-CCP for $L.$ Let $\lambda $ be a law of chance of L with chance hypothesis $\Lambda = L^{-1}(\lambda ).$ Since L is a theory of chance, we have $\lambda (\Lambda )>0.$ Since $\mu $ merges with $\lambda ,$ Proposition 5.7 then implies that $\mu (\Lambda )>0.$ So $\mu $ must assign positive probability to each chance hypothesis of L—so L can have only countably many chance laws.

Corollary 5.9. Super-Determinism and Basic Frequentism are M-CCP-unsatisfiable (and hence also M-CCCP-unsatisfiable).

As an immediate consequence of Remark 5.5 and Propositions 3.4 and 5.6 we have:

Proposition 5.10. If a prior satisfies CCP for a theory of chance, then it also satisfies M-CCP for that theory.

Corollary 5.11. A theory is M-CCP satisfiable if it is CCP satisfiable. In particular, Computable Super-Determinism, Computable Frequentism, and the Toy Best System Theory are M-CCP-satisfiable.

Remark 5.12 (Mixtures and Merging).

Let $\{ \mu _k \}_{k\in I}$ be a countable family of probability measures at least one of which is a prior and for each $k \in I,$ let $q_k>0$ with $\sum _{k\in I} q_k=1.$ Then $\mu : = \sum q_k \cdot \mu _k$ is a prior. And $\mu $ merges with any measure merged with by one of the $\mu _k$ : if $\nu $ is merged with by $\mu _k,$ then (by Proposition 5.7), so (given how $\mu $ was defined), and so $\mu $ merges with $\nu $ (by Proposition 5.6).

In the case of Computable Super-Determinism, a prior satisfies CCP if and only if it satisfies M-CCP. But, for more typical theories, we expect that there will be priors that satisfy M-CCP without also satisfying CCP.

Example 5.13. A theory L and a prior $\mu $ may satisfy M-CCP without satisfying CCP (even if L is proper and $\mu $ is computable): let L be the theory that assigns the fair coin measure $\nu _{.5}$ to every sequence and let $\mu ^\dagger $ be the prior that thinks there is a .9 chance that the first bit is a 0 and thinks that each subsequent bit is chosen by flipping a fair coin. We have merging (because $\nu _{.5}$ and $\mu ^\dagger $ agree completely when conditionalized on any non-trivial initial segment) but CCP is violated.

Example 5.14. Improper theories of chance are never CCP-satisfiable. But they can be M-CCCP satisfiable. Consider the theory of chance that assigns the measure $\mu ^\dagger $ of the preceding example to each sequence that begins with 0 as its law of chance (and which considers sequences that begin with 1 to be lawless). This theory is CCP-unsatisfiable (because it is improper— $\mu ^\dagger $ assigns its chance hypothesis probability .9). But the fair coin measure (considered as a prior) satisfies M-CCP for this theory.

We have a partial converse to Corollary 5.11.

Proposition 5.15. Any proper theory of chance that is M-CCP-satisfiable is also CCP-satisfiable.

Proof. Let L be a proper theory of chance that is also M-CCP satisfiable. We know from Proposition 5.8 that L has a countable family of laws of chance: $\lambda _1$ , $\lambda _2,$ …. Let $\mu $ be a prior that merges with each $\lambda _k.$ A result of Ryabko tells us that if $\mathcal {K}$ is a set of probability measures on $\mathsf {2}^\omega $ and there exists a probability measure $\mu $ that merges with each measure in $\mathcal {K},$ then there exists a measure $\mu ^*$ that merges with each measure in $\mathcal {K}$ and which can be written as a sum of measures, each of which is a positive multiple of a measure in $\mathcal {K}.$ ⁵⁸ Applied to our case: there must be a $\mu ^*$ that is a weighted sum of the $\lambda _k$ and which merges with each $\lambda _k.$ By Proposition 5.7, each —so each $\lambda _k$ appears with positive weight in $\mu ^*.$ So $\mu ^*$ (or the result of adding something to it to make sure that we have a prior as in the proof of Corollary 3.6) satisfies CCP for L (by Proposition 3.4).

Of course, it does not follow from this that M-CCCP-satisfiability implies CCCP-satisfiability for a proper theory of chance L: it could be that the only computable priors that merge with every chance law of L are not themselves weighted sums of the chance laws of $L.$ So, in particular, the above results leave open the questions whether Computable Super-Determinism, Computable Frequentism, and the Toy Best System Theory are M-CCCP-satisfiable. We will see in Proposition 5.25 that Computable Super-Determinism is M-CCCP-unsatisfiable. We can use a variant of the proof of Proposition 4.9 to show that there are no computable priors that satisfy M-CCP for Computable Frequentism or for the Toy Best System.

Proposition 5.16. Let $R=\{ r_k\}_{k\in J}$ be a set of computable binary sequences and for each $k \in J,$ let $\nu _k$ be the Bernoulli measure whose parameter admits the binary expansion $r_k.$ If $\mu $ is a computable prior that merges with each $\nu _k,$ then there is an extrapolating machine that NV-learns each $r_k.$

Proof. It suffices to show that for each $r_k \in R$ , $\mathsf {ML}_\mu \cap \mathsf {ML}_{\nu _k} \neq \varnothing ,$ and then to argue as in steps (2) and (3) of the proof of Proposition 4.9.

For each $k \in J,$ let $N_k$ be the set of sequences in which 0’s have limiting relative frequency $r_k.$ Then we can write $\mu = \nu + \sum _{k\in J} c_k \mu _k,$ where $\nu $ is a (possibly trivial measure) that considers each $N_k$ a null set, each $c_k>0,$ and each $\mu _k$ is a probability measure that assigns $N_k$ probability one (so $c_k \cdot \mu _k$ is the restriction of $\mu $ to $N_k$ ). Since $\mu $ merges with each $\nu _k,$ we know via Proposition 5.7 that from which it follows that

The $\mu _k$ may or may not be computable. But the notion of Martin-Löf randomness can be generalized to apply to arbitrary probability measures: we call a sequence S blindly Martin-Löf random relative to the probability measure $\mu $ if there is no uniformly effective family of open sets $U_1$ , $U_2,$ … such that $S \in \bigcap U_k$ and $\mu (U_k)\leq 2^{-k}$ for each $k.$ ⁵⁹ We write $\mathsf {BML}_\mu $ to denote the set of sequences blindly Martin-Löf random relative to the probability measure $\mu $ (of course $\mathsf {BML}_\mu = \mathsf {ML}_\mu $ when $\mu $ is computable). Note that we have $\mu (\mathsf {BML}_\mu )=1$ for any probability measure $\mu .$ ⁶⁰

We claim that for any $k \in J,$ we have $\mathsf {BML}_{\mu _k} \cap \mathsf {ML}_{\nu _{k}} \neq \varnothing .$ For suppose otherwise. Then of course $\mu _k(\mathsf {ML}_{\nu _{k}})=0$ (since $\mu _k (\mathsf {BML}_{\mu _k})=1$ ). But since $\nu _k(\mathsf {ML}_{\nu _k})=1,$ this contradicts the fact that

And we can now argue, as in the first step of the proof of Proposition 4.9, that since $\mu $ can be decomposed as a sum of measures in which each $\mu _k$ appears with non-zero weight, every sequence in $\mathsf {BML}_{\mu _k}$ must also be in $\mathsf {ML}_{\mu }.$ So we have, as desired, that for each $r_k \in R$ , $\mathsf {ML}_\mu \cap \mathsf {ML}_{\nu _k} \neq \varnothing .$

Corollary 5.17. There are no computable priors that satisfy M-CCP for Computable Frequentism or for the Toy Best System.

5.2 Weak merging

M-CCP is weaker than CCP and M-CCCP is weaker than CCCP. But for proper theories of chance M-CCP satisfiability is equivalent to CCP-satisfiability and it would be surprising if there were a significant gap between M-CCCP-satisfiability and CCCP-satisfiability for such theories. So it makes sense to consider a further alternative to CCP. Here is another criterion of learning that has played some role in game theory. ⁶¹

Definition 5.18 (Weak merging).

Let $\lambda $ and $\mu $ be probability measures on $\mathsf {2}^\omega .$ We say that $\mu $ weakly merges with $\lambda $ if there is a set of sequences of $\lambda $ -measure one such that for each S in that set:

$$ \begin{align*} \lim_{n \to \infty} | \mu(0 \, | \,S{\upharpoonright}n) - \lambda(0\, | \, S{\upharpoonright}n) | = 0. \end{align*} $$

Definition 5.19 (WM-CCP).

Let L be a theory of chance. We say that a prior $\mu $ on $\mathsf {2}^\omega $ satisfies the Weak Merging Chance–Credence Principle (WM-CCP) for L if $\mu $ weakly merges with each chance law of $L.$

If a prior $\mu $ weakly merges with a proper law of chance $\lambda $ of a theory of chance, then we can take the set of sequences that witnesses weak merging to be a subset of the chance hypothesis of $\lambda .$

Definition 5.20. We call a theory of chance L WM-CCP-satisfiable if there is a prior that satisfies WM-CCP for $L.$ If there is a computable prior with this feature, we say that L is WM-CCCP-satisfiable.

The following is immediate.

Proposition 5.21. If one probability measure merges with another, then the first also weakly merges with the second. If a theory of chance is CCP- or M-CCP-satisfied (CCCP- or M-CCCP-satisfied) by a given prior, then it is also WM-CCP-satisfied (WM-CCCP-satisfied) by it. If a theory is CCP- or M-CCP-satisfiable (CCCP- or M-CCCP-satisfiable) then it is also WM-CCCP-satisfiable (WM-CCCP-satisfiable).

Remark 5.22 (Stronger than it looks).

In moving from merging to weak merging we lower our sights—rather than requiring that, in the limit of large data sets sampled from $\lambda ,$ for every event, the judgement of $\mu $ of the probability of that event must conform to the judgement of $\lambda $ (in a fashion uniform across events) we require this only for a special class of events—the identity of the next bit to be revealed.

But to require weak merging is to require more than may be immediately apparent. It turns out that $\mu $ weakly merges with $\nu $ if and only if: as the size of the data set goes to infinity, for each k, $\mu $ must asymptotically learn to defer to $\nu $ in a way uniform across all events that look k time-steps beyond the data about the chance of all such events. ⁶² So merging is stronger than weak merging precisely in paying attention to events involving an infinite number of times (a type events about whose epistemological salience a Humean might well be suspicious).

Example 5.23. Consider the Bayes–Laplace prior. ⁶³ This is the computable probability measure $\bar {\nu }$ determined by the rule that if $\tau $ is an m-bit string containing $\ell $ 0’s, then

$$ \begin{align*} \bar{\nu} ([\![ \tau ]\!]) := \frac{\ell! (m-\ell)!}{(1+ m)!}. \end{align*} $$

Famously, there is a sense in which $\bar {\nu }$ is a Lebesgue-uniform mixture of the Bernoulli measures—and from this it follows that $\bar {\nu }$ weakly merges with each Bernoulli measure. ⁶⁴ It then follows that $\bar {\nu }$ satisfies WM-CCP (indeed, WM-CCCP) for Basic Frequentism and for Computable Frequentism.

This example shows that theories with uncountably many laws of chance can be WM-CCP-satisfiable (and even WM-CCCP-satisfiable), even though they must be CCCP-, CCP-, M-CCCP, and M-CCP-unsatisfiable. But there are limits.

Proposition 5.24. Let L be a theory of chance and let $\mu $ be a prior that satisfies WM-CCP for $L.$ Then the laws of chance of L include only countably many delta-functions. In particular, Super-Determinism is WM-CCP-unsatisfiable.

Proof. We generalize the notion of an extrapolating machine: an extrapolator is a (not necessarily computable) function from binary strings to bits. We extend the notion of NV-learning to extrapolators: extrapolator m NV-learns sequence S if when shown S bit by bit, m is eventually always correct in its predictions about the next bit. The set of sequences NV-learned by an extrapolator m is always countable (each such sequence can be generated by extending some binary string by asking m what it expects to see next). Any prior $\mu $ determines an extrapolator $m_\mu ^*$ via the rule: $m_\mu ^*(\sigma )=0$ if $\mu (0\, | \, \sigma )\geq .5,$ otherwise $m_\mu ^*(\sigma )=1.$

Suppose that prior $\mu $ satisfies WM-CCP for theory of chance $L.$ Suppose that for some $S \in \mathsf {2}^\omega $ , $\delta _S$ is a law of chance of $L.$ We can make $\mu ( 0 \, | \, S{\upharpoonright }n)$ as close as we like to $\delta _S(0 \, |\, S{\upharpoonright }n)$ by choosing n sufficiently large. It follows that $m_\mu ^*$ NV-learns $S.$ Since $m_\mu ^*$ cannot NV-learn an uncountable set of sequences, there can only be countably many delta-functions among the laws of chance of $L.$

Computable Super-Determinism and the Toy Best System Theory are WM-CCP-satisfiable (being CCP-satisfiable). But they are not WM-CCCP-satisfiable.

Proposition 5.25. No computable prior weakly merges with each delta-function concentrated on a computable sequence.

Proof. Let $\mu $ be a computable prior with associated extrapolating machine $m_\mu $ (as in Section 4.1) and let S be a binary sequence. Arguing as in Proposition 5.24 we see that if $\mu $ weakly merges with $\delta _S,$ then $m_\mu $ NV-learns $S.$ But Proposition 4.5 implies that no extrapolating machine NV-learns each computable sequence.

Remark 5.26 (Mixtures and weak merging).

In Remark 5.12, we saw that when we build a prior by taking a non-trivial mixture of some measures, that prior merges with every probability measure merged with by any component of the mixture. The picture is different with weak merging: Ryabko and Hutter give an example in which $\mu $ is a prior, $\nu $ is a probability measure, and $\frac {1}{2}( \mu + \nu )$ fails to weakly merge with a delta-function measure that is weakly merged with by $\mu .$ ⁶⁵ So it is not obvious that you can build a prior suited to a Best System theory of chance by taking a mixture of a prior (such as the Bayes–Laplace prior) that weakly merges with a desirable range of Bernoulli measures with a measure that weakly merges with a desirable range of delta-function measures.

6 Scorecard

• Super-Determinism: Because this theory has uncountably many chance laws, it is CCP-unsatisfiable (Proposition 3.4) and M-CCP-unsatisfiable (Corollary 5.9). A learning-theoretic argument shows that it is also WM-CCP-unsatisfiable (Proposition 5.24). It is, then, of course also CCCP-unsatisfiable, M-CCCP-unsatisfiable, and WM-CCCP-unsatisfiable.

• Computable Super-Determinism: For this theory, any prior that can be written as a sum in which each delta-function measure concentrated on a computable sequence appears with non-zero weight will satisfy CCP (Proposition 3.4); it will therefore also satisfy M-CCP (Corollary 5.11) and WM-CCP (Proposition 5.21). However, it turns out that no such prior is computable—and, indeed, that the theory is CCCP-unsatisfiable (Proposition 4.7). It is WM-CCCP-satisfiable (Proposition 5.25) and therefore (via Proposition 5.21) also M-CCCP-unsatisfiable.

• Basic Frequentism: Because this theory has uncountably many chance laws, it is CCP-unsatisfiable (Proposition 3.4) and M-CCP-unsatisfiable (Corollary 5.9). It is of course therefore also CCCP-unsatisfiable and M-CCCP-unsatisfiable. But it is WM-CCP-satisfiable and even WM-CCCP-satisfiable (Example 5.23).

• Computable Frequentism: For this theory, any prior that can be written as a sum in which each Bernoulli measure $\nu _r$ with a computable parameter r appears with non-zero weight will satisfy CCP (Proposition 3.4). Priors of this form will also satisfy M-CCP (Proposition 5.10) and WM-CCP (Proposition 5.21). But since no prior that can be written as a sum in which each computable Bernoulli measure appears with non-zero weight can be computable (Corollary 4.10), Computable Frequentism is not CCCP-satisfiable (by Proposition 3.4). Nor is it computably M-CCP satisfiable (see Corollary 5.17). On the other hand: Computable Frequentism is WM-CCCP-satisfiable (Example 5.23).

• Toy Best System Theory: The Toy Best System is not CCCP-satisfiable (see Proposition 4.7 or Corollary 4.10). It is also not WM-CCCP-satisfiable (Proposition 5.25)—from which it follows that is not M-CCCP-satisfiable (Proposition 5.21—or see Corollary 5.17). On the other hand, this theory is CCP-satisfiable (Proposition 3.4). So it is also M-CCP-satisfiable (Corollary 5.11) and WM-CCP-satisfiable (Proposition 5.21).

7 Conclusion

The main goal of this paper has been to show that it is more difficult one might think to find a consistent package consisting of a reductionist theory of chance and a principle encoding the rational relation between credence and chance—especially if one expects there to be rationally permitted computable priors. ⁶⁶

In Section 3 we saw there exists a prior probability measure satisfying CCP relative to a given proper theory of chance if and only if that theory has only countably many chance laws. This is, of course, a direct consequence of the choice to use ordinary (i.e., countably additive) probability measures to model rational credal states. Some reductionists about chance may be happy to ignore theories with uncountably many chance laws—e.g., advocates of best-system approaches who take fully seriously the idea that each chance law should be something like a humanly expressible theory of the world. But many reductionists will, I suspect, regard Basic Frequentism, the laws of chance of which are the Bernoulli measures, as a theory that should be compatible with our account of the relation between chance and credence. In the first instance, such reductionists face a choice between accepting that credal states can be represented by objects more general than probability measures and replacing CCP with something weak enough to accommodate the existence of rationally permitted priors adapted to theories of chance with uncountably many chance laws.

In Section 4 we saw that even if priors exist that satisfy CCP relative to a given theory of chance, there may be no computable priors with this feature. In particular this happens when the laws of chance of a theory include each delta-function measure concentrated on a computable sequence—the root of the problem being that the computable sequences, though enumerable, are not computably enumerable. A related problem arises for theories whose chance laws include each Bernoulli measure with a computable parameter. I do not think we should rest content with an account of chance and credence that tells us that rationality requires us to adopt a non-computable credence function—any more than we would be satisfied with an account of rationality that required rational agents to be able to solve the Halting Problem. Some reductionists may be happy with the moral that certain prima facie attractive theories of chance are unsuited to computable agents—such reductionists will then need to be careful to avoid using computationally intractable sub-families of the delta-function measures and Bernoulli measures as laws of chance. Others may again be interested in investigating whether the problem can be avoided by generalizing the Bayesian framework or by endorsing a constraint of theories of chance on rationally permitted priors weaker than CCP.

There are a number of standard strategies that Bayesians have explored in other contexts in which the presence of an uncountable number of alternatives and/or awkward null sets causes trouble, including infinitesimal-valued credence functions, merely finitely additive probabilities, regular conditional probabilities, and primitive conditional probabilities. These are considered briefly in Appendix B, where it is argued that none of them offers much help in our current predicament.

Section 5 explored two ways of weakening CCP with an eye to satisfying those desiderata: M-CCP and WM-CCP. The latter turns out to be the more fruitful for our purposes. It says, roughly, that the rational priors are those with the feature that, almost certainly, in the limit of large data sets the posterior probabilities will converge to the true chances for events that depend on only finitely many times. WM-CCP has the attractive feature that it is satisfied by the computable Laplace–Bayes prior, both relative to Basic Frequentism (with its uncountable family of chance laws) and relative to Computable Frequentism (with its enumerable but computationally intractable family of chance laws). So it is a substantive constraint on priors that is consistent with the intuition that frequentism in its several forms is one of the more forgiving of reductionist approaches—and with the thesis that rationality should be consistent with computability. But like CCP, WM-CCP cannot be satisfied by any computable prior relative to a theory of chance that includes among its laws of chance all delta-function measures concentrated on computable sequences.

Other responses to the problems encountered in Sections 3 and 4 are of course possible. One of the most natural would be to reject the assumption that we have implicitly carried over from the literature on the Principal Principle: that our account of the relation between credence and chance should be the basis of a dichotomy between priors that are rationally permitted and those that are not rationally permitted. Perhaps we should instead think of considerations about chance as inducing a partial ordering more rational than on possible priors. For instance, given a theory of chance L and a prior $\mu $ we could look for the largest sub-theory of L (= restriction of L to a subset of its domain of definition) such that $\mu $ satisfies one of our principles (CCP, M-CCP, or WM-CCP) for that sub-theory, considering $\mu _1$ to be no more rational than $\mu _2$ if the sub-theory $L_1$ of L associated with $\mu _1$ is itself a sub-theory of the sub-theory $L_2$ or L associated with $\mu _2.$ If there are priors that satisfy the given principle for a given theory of chance then they will be fully rational in the sense that no prior is more rational then them—but otherwise, there will just more and less rational priors without any being fully rational.

Applied to CCP or to M-CCP, this strategy is not particularly enticing: it tells us that the fair-coin measure is a more rational prior for Basic Frequentism than the Laplace–Bayes indifference prior—even though agents with the fair coin measure as their prior will remain certain the chance that the next bit will be 0 is .5 no matter what data they see, while the posterior probabilities that agents with the Laplace–Bayes prior assign to that event will converge (almost surely) to the true chance (if one there be) in the limit of large data sets. But applied to WM-CCP, this strategy suggests a way forward for fans of the best-system-type approaches in grappling with the problems surrounding delta-function measures concentrated on computable sequences (although some challenges would remain).

This is far from a complete account of possible responses to problems exposed above. But I hope, at least, that one moral is clear: interesting work remains to be done for anyone who is attracted to Lewis’s accounts of chance and credence and who is also inclined to take the computability of priors to be consistent with their rationality.