2. Mathematical framework

A binary sequence is a map $S:{\mathbb{N}^ + } \to \left\{ {0,1} \right\}$ . We denote by ${2^\omega }$ the set of all binary sequences. For any binary sequence $S$ and natural number $n$ , we write $S{\upharpoonright} n$ for the binary string consisting of the first $n$ bits of $S$ . If $\tau $ is a binary string, then we denote by ${[[}{ \tau} {]]}$ the set of all binary sequences whose initial bits are given by $\tau $ . We call such ${[[} \tau {]]}$ the basic subsets of ${2^\omega }$ .

We will call a subset of ${2^\omega }$ an event if it can be constructed out of basic subsets by taking complements and unions a countable number of times in any order. ^{Footnote 7} A probability measure on ${2^\omega }$ is a map $\nu $ that assigns numbers between 0 and 1 (inclusive) to events, that assigns 1 to ${2^\omega }$ , and that is countably additive (i.e., if ${E_1},{E_2}, \ldots $ are pairwise disjoint events, then $\nu \left( { \cup {E_k}} \right) = \sum \nu \left( {{E_k}} \right)$ ). Any such map is completely determined by its behavior on basic subsets.

A binary sequence $S$ is computable if there is a Turing machine that on input of any $n \in \mathbb{N}$ gives as output $S{\upharpoonright}n$ . A real number $x$ is computable if there exists a Turing machine that on input $n$ returns a rational number within $1/n$ of $x$ . A probability measure $\nu $ on ${2^\omega }$ is computable if there exists a Turing machine that on input of $n$ and binary string $\tau $ returns a rational number within $1/n$ of $\nu ( [[{ \tau ]]})$ .

If $S$ is a binary sequence then the Dirac measure ${\delta _S}$ concentrated on $S$ is the probability measure that assigns probability 1 to event $E$ if $S \in E$ and assigns probability 0 to $E$ otherwise. A Dirac measure is computable if and only if the sequence it is concentrated on is.

If $r$ is a number (strictly) between zero and one, then the Bernoulli measure ${\nu _r}$ with parameter $r$ is the probability measure that, for any binary string $\tau $ containing $\ell $ 0s and $m$ 1s, assigns the basic subset $[[{ \tau} ]]$ probability ${r^\ell }{(1 - r)^m}$ . We call ${\nu _{.5}}$ the fair coin measure. A Bernoulli measure is computable if and only if its parameter is.

3. Worlds

We will be concerned with simple worlds whose histories can be encoded in binary sequences: At each such world, time has the structure of the natural numbers and the history of such a world just tells us, for each time, whether or not a single property is instantiated at that time. We work with these worlds in order to have a precise and tractable framework in which to interpret Lewis’s ideas about chance and credence. The lessons we learn can be translated to richer settings (our “possible worlds” could always be interpreted as subsystems of more interesting worlds).

A probability measure $\nu $ on ${2^\omega }$ can be thought of as an instruction manual for building a world. Nature is equipped with a set of coins with each possible bias. A probability measure $\nu $ on ${2^\omega }$ tells Nature to flip a coin with bias $\nu \left( {[[} 0{]]} \right)$ to determine whether the first bit is a 0. More generally, $\nu $ tells nature to flip a coin of bias $\nu ({[[}{\tau 0}{]]}|{[[} {\tau {]]}})$ to determine whether the next bit will be 0, given that $\tau $ gives the history so far. ^{Footnote 8} A Dirac measure ${\delta _S}$ instructs Nature to construct a history by flipping coins of maximal bias in a way guaranteed to generate $S$ . A Bernoulli measure ${\nu _r}$ tells Nature to generate each bit by flipping a coin of bias $r$ .

4. Chance and credence

Reductionism about chance is the thesis that chance-y facts supervene on non-chance-y facts. So in our context, a reductionist account of chance can be encoded in a map from ${2^\omega }$ to the space of probability measures over ${2^\omega }$ (this map will be merely partially defined if there are lawless worlds without chance facts). We call a probability measure $\lambda $ a chance law of such an account if the relevant map assigns it to some world.

For present purposes, a Bayesian prior is a probability measure on ${2^\omega }$ that assigns positive probability to each basic subset of ${2^\omega }$ . ^{Footnote 9}

It has seemed to many that rationality requires credence to defer to chance in a certain sense: In some situations, if you are rational, then your credence in an event must coincide with the chance of that event.

Fix a reductionist account of chance (such as the Best System Account). Corresponding to any chance law $\lambda $ of the theory, there is the set of worlds ${\rm{\Lambda }}$ at which $\lambda $ gives the chance facts. We call this set the chance hypothesis corresponding to that law. This is the chancemaking proposition: the proposition that the chance facts are given by $\lambda $ .

The requirement that rational priors should satisfy the Chance-Credence Principle for a given reductionist theory of chance is a weak form of Lewis’s Principal Principle. ^{Footnote 10}

A prior $\mu $ satisfies the Chance-Credence Principle for a given reductionist theory of chance if and only if: (i) the theory of chance admits only countably many chance laws ${\lambda _1},{\lambda _2}, \ldots $ (with chance hypotheses ${{\rm{\Lambda }}_1},{{\rm{\Lambda }}_2}, \ldots $ ); (ii) each chance law ${\lambda _k}$ of the theory is proper, in the sense that ${\lambda _k}\left( {{{\rm{\Lambda }}_k}} \right) = 1$ ; and (iii) $\mu $ can be written as a weighted sum of probability measures in which each ${\lambda _k}$ appears with non-zero weight (and in which any other summands are concentrated on the set of lawless worlds).

5. The Best System Account

Lewis (Reference Lewis1994, ${\rm{\S}}$ 4) tells us a bit about the map that encodes his favored reductionist theory of chance. He works in the context of finite worlds that can be encoded in binary strings. In this context, under a straightforward frequentist approach a world $\sigma $ would be assigned a Bernoulli measure ${\nu _r}$ as its chance law if and only if $r$ gives the relative frequency of 0s at $\sigma $ . Lewis emphasizes that his Best System Account departs in two ways from this sort of frequentism.

• Some worlds will not be assigned a Bernoulli measure as their Best-System chance law: some worlds exhibit patterns that render it natural to think of Nature as generating bits by following instructions that call for a coin of bias ${r_1}$ to be flipped to generate the first bit; a coin of bias ${r_2}$ to be flipped to generate the second bit; and so on. As an extreme case, Lewis mentions a world where history alternates between 0 and 1, which is naturally thought of as generated by alternating between flipping coins with maximal bias in favor of 0 and 1—in other words, the chance facts there are encoded in a Dirac measure.

• On the other hand, Lewis thinks that in some cases the chance law at a world should be a Bernoulli measure ${\nu _r}$ even though $r$ does not give the relative frequency of 0s at that world (e.g., in a case where the relative frequency of 0s at a world is given by a messy number ${r^{\rm{*}}}$ close to $.5$ then the fair coin measure may provide a better candidate than ${\nu _{{r^{\rm{*}}}}}$ to be the Best-System chance law of the world—although of course for long-lived worlds, the chance of getting a relative frequency of 0s differing even a little from $.5$ by tossing a fair coin will be minuscule).

How does this look when transposed to the setting of worlds whose histories are coded in infinite binary sequences? It is helpful to note two important features of the infinite context.

• The idea behind the Best System Approach is that the chance facts at a world are given by the best scientific summary of that world—where best means something like best by our ordinary scientific standards. It is crucial to this picture, then, that the systems in competition be finitarily specifiable objects—otherwise, the question of which of them is best according to ordinary scientific standards loses all sense. The specification of an arbitrary real number is an infinitary task—requiring, in general, the specification of infinitely many bits. So the specification of an arbitrary probability measure on ${2^\omega }$ is infinitary twice over—requiring, in general, the assignment of a real number to each of the infinitely many binary strings. Now, as Turing (Reference Turing1936, 236) tells us: “The ‘computable’ numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means.” In the same way, the finitarily specifiable probability measures are the computable probability measures. So in the infinite context, only computable measures should be eligible candidates to be chance laws under the Best System Approach.

• In this setting there is a natural way to make precise the notion of a random-looking sequence: the notion of a Martin–Löf random sequence. ^{Footnote 11} If $S$ is a binary sequence and $\mu $ is a computable probability measure, then, roughly speaking, $S$ is $\mu $ -Martin–Löf random just in the case that $S$ exhibits no finitarily specifiable patterns of behavior that would be arbitrarily surprising to an agent expecting to see a data stream generated by $\mu $ . ^{Footnote 12}

I claim that any plausible version of the Best System Analysis should conform to the following constraints.

6. Trouble

BS1–BS3 and the Chance-Credence Principle are mutually consistent. Consider the version of the Best System Account in which each computable sequence is assigned the corresponding Dirac measure as its chance law, each non-computable sequence is assigned a computable Bernoulli measure as its chance law if and only if it is Martin–Löf random with respect to it, and all other sequences are lawless. Then any prior that can be written as a (non-trivial) weighted sum of each of the countably many computable Dirac measures and computable Bernoulli measures will satisfy the Chance-Credence Principle.

But no computable prior can satisfy the Chance-Credence Principle for any reductionist account of chance obeying BS1 and BS2. As noted above, BS1 and BS2 imply that each computable sequence $S$ is assigned a chance law that assigns $\left\{ S \right\}$ positive probability. It follows that if a prior $\mu $ satisfies the Chance-Credence Principle for an account of chance obeying BS1 and BS2, then it must be possible to write $\mu $ as a weighted sum of probability measures in which each computable Dirac measure appears with positive weight. We will show that no such prior can be computable. ^{Footnote 17} Recall that $S{\upharpoonright}k$ is the binary string consisting of the first $k$ bits of $S$ . So $\mu ({[[} S{\upharpoonright}\left( {n + 1} \right) {]]}|{[[}S {{\upharpoonright}n} {]]})$ is the probability that $\mu $ gives to seeing the next bit of $S$ after having been shown the first $n$ bits of $S$ . Let us say that prior $\mu $ learns sequence $S$ if, after seeing sufficiently long initial segments of $S$ , $\mu $ is always able to correctly predict the next bit with credence exceeding some fixed cut-off. ^{Footnote 18} Consider any prior $\mu $ that can be written in the form $\mu : = \nu + c \cdot {\delta _S}$ , where $S$ is a binary sequence, $c \gt 0$ , and $\nu $ is a measure that considers $\left\{ S \right\}$ a nullset. Since $\nu \left( {\left\{ S \right\}} \right) = 0$ , we can make $\nu \left( {[[} {S{\upharpoonright}n} {]]} \right)$ —and hence also $\nu \left( {[[} {S{\upharpoonright}\left( {n + 1} \right)} {]]}\right)$ —as small as we like by choosing $n$ sufficiently large. ^{Footnote 19} It follows that $\mu $ learns $S$ . So, in particular, if $\mu $ satisfies the Chance-Credence Principle for a Best System Account obeying BS1 and BS2, then $\mu $ must learn each computable sequence. But given any computable prior $\mu $ , we can construct a computable binary sequence $S_\mu ^\ast $ that $\mu $ does not learn: $S_\mu ^\ast $ consists of blocks of 0s separated from each other by individual 1s; a 1 is called for whenever $\mu $ has just seen at least one 0 and thinks that it is more likely than not that the next bit will be a 0. So no computable prior can satisfy the Chance-Credence Principle for a reductionist account of chance obeying BS1 and BS1.

A variant of this argument shows that no computable probability measure can be written as a weighted sum of probability measures in which each computable Bernoulli measure appears with non-zero weight—from which it follows that no computable prior can satisfy the Chance-Credence Principle for any reductionist account of chance obeying BS3.

7. Options

Roughly speaking, the problem that we have run into is that 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 implies that if you are rational, then to the extent that you are confident in a chance hypothesis it should guide your estimates of probability. 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 law at your world, if you are a good scientist and you see enough data, you should become confident in something like that law and so you should eventually mimic it in your estimates of probability. As Lewis (Reference Lewis1986, 121) puts it: “if we start with a reasonable initial credence function and do enough feasible investigation, we may expect our credences to converge to the chances.” But this is a setting in which no computable universal learner can exist.

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. So I think we ought to be interested in revising some part of the framework used above, rejecting one or more of BS1–BS3, or weakening the Chance-Credence Principle. I end with a few observations about these options.

1. (1) Would countenancing rational credal states represented by merely finitely additive probability measures help? No. For suppose that a prior $M$ of this kind exists such that: (i) for any computable $S$ , $M\left( {\left\{ S \right\}} \right) \gt 0$ ; and (ii) $M\left( {[[}\tau {]]} \right)$ is defined for each binary string $\tau $ . Then there must exist a countably additive $\mu $ such that $\mu \left( {[[} \tau {]]} \right) = M\left( {[[} \tau {]]} \right)$ for each string $\tau $ , and such that $\mu \left( {\left\{ S \right\}} \right) \gt 0$ for each computable sequence $S$ . ^{Footnote 20} Our diagonalization argument above then tells us that such an $M$ cannot be computable even in the sense that its restriction to basic sets is computable.

2. (2) Our Chance-Credence Principle requires that each law of chance $\lambda $ be proper, in the sense that it assigns probability one to its own chance hypothesis ${\rm{\Lambda }}$ . We could relax this restriction to allow $\lambda \left( {\rm{\Lambda }} \right) \in \left( {0,1} \right]$ . In that case it would be natural to replace the Chance-Credence Principle by the following. ^{Footnote 21}

The moral is: 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 computability to be a constraint on rational priors.