1. Introduction

The idea that determinism does not rule out the existence of objective or physical probabilities is quite old, at least in physics. Nowadays, this idea can be made precise. Determinism is consistent with probabilistic statements if the interpretation of probability is the frequency interpretation.Footnote ¹

While frequentists need not worry about determinism, they should worry about the practical relevance of frequentist probabilities. According to the most widely held version of the frequency theory, frequentist probability information is information about long-run properties of infinite sequences of events.Footnote ² Why should this information be relevant to a decision maker, who wants to predict events in a finite number of cases? This is usually called the problem of the single case, but since any finite number of cases raises the same problem, finite case is more fitting.

The problem of the finite case is a serious one. The history of probability theory demonstrates that one important reason for the interest in objective probabilities is that they are assumed to provide a basis for decision making. Even in the natural sciences, where the use of objective probabilities is motivated by other concerns, decision making becomes important once probabilistic hypotheses are subjected to statistical tests. A physicist's interest in the probabilities of quantum mechanics may be purely theoretical. Nevertheless, it is unlikely that this interest would exist if not at least some probabilistic hypotheses—like the hypothesis of radium decay—were testable, which presupposes that, in principle, the probabilities are relevant for (statistical) decision making.Footnote ³ Arguably, then, theories of objective probability implying the irrelevance of objective probabilities for decision making miss the point because such probabilities cannot play the role they have been assigned in practice, including in scientific practice.

Thus, the frequency theory as a theory of practically relevant probabilities is in difficulties.Footnote ⁴ However, help has been offered from unlikely quarters. Howson and Urbach (Reference Howson and Urbach1993, Chapter 13) argue that frequentist probabilities are relevant for finite-case decision making, if only for Bayesians.

In this paper, I argue that Howson and Urbach's, or any other, Bayesian solution to the problem of the finite case cannot work. The features of the frequency theory that allow for nondegenerate probabilities in a deterministic world also imply the irrelevance of these probabilities for decision making.

The paper proceeds as follows. Section 2 presents the structure of the argument. Section 3 reviews some facts concerning frequentist probability theory and its relation to deterministic dynamics. Section 4 then uses these facts to discuss the claims of Howson and Urbach and the possibility of forging a Bayesian connection between relative frequencies and subjective probabilities. Section 5 concludes.

2. The Structure of the Argument

The frequency theory defines the probability of an event as its limiting relative frequency in an infinite random sequence generated by repetitions of some experiment (Howson and Urbach Reference Howson and Urbach1993, Chapter 13). There are several definitions of randomness for infinite sequences but, again, all of them involve only long-run properties. This gives rise to the following argument.

The premise LR (for “long run”) implies that the frequency theory is completely silent on how the required long-run properties of sequences come about. It is possible to assume, then, that each single event in a random sequence is determined in accordance with deterministic laws. In fact, sequences with any required long-run properties can be generated by simple deterministic systems (see Section 3). Thus, LR implies $C_{1}$.

The consistency of the frequency theory with determinism is often viewed as a point in its favor. But this favorable result derives from the same premise as the unfavorable consequence that frequentist probability statements are irrelevant for decision making. Since any finite part of a sequence is irrelevant for the sequence's long-run properties, LR implies the following intermediate result IR:

Intuitively, IR implies $C_{2}$. If this is correct, frequentist-probability information is irrelevant for Bayesian decision making.

The intuition that IR implies $C_{2}$ is in need of a proof. Frequentist probability information actually constrains subjective probability measures on an appropriately specified set of infinite sequences. However, according to $C_{2}$, these constraints have no implications for the probabilities of finite segments. This is plausible, but not obvious. Even if frequentist-probability information rules out no finite segment as impossible, it might imply that some segments are less probable than others under any consistent assignment of subjective probabilities. However, this is not the case. In order to understand why, we have to look at frequentist probability theory in some detail (see Section 3).

Given this background, it is easy to derive $C_{2}$. Section 4 considers the set of infinite sequences introduced in Section 3 and shows that, on this set, there always exists a subjective prior probability measure with the following two properties: (i) the measure is consistent with the given frequentist-probability information; (ii) it is consistent with arbitrary subjective probability assignments in the finite case. Moreover, the result covers the different versions of the frequency theory deriving from different definitions of randomness.

The next crucial issue is whether subjectivists might appeal to supplementary principles in order to overcome $C_{2}$. Of course, on a personal level, subjectivists can accept any constraints on the choice of their prior probabilities. This, however, is not at issue here. Subjective Bayesianism is a theory of rational beliefs. The question is whether subjectivists are obliged to heed any rationality constraints (Howson and Urbach Reference Howson and Urbach1993, 72) on the choice of their priors, and whether such constraints might contradict $C_{2}$.

The regularity principle, which requires that all possibilities must get positive probability,Footnote ⁵ might be a rationality constraint because a nonregular prior can make it impossible to learn the truth by assigning it zero probability. However, the regularity principle cannot topple $C_{2}$.

A widely accepted principle that would do the job is the Principal Principle (PP). (See Lewis [Reference Lewis1980] Reference Lewis1986, and Howson and Urbach Reference Howson and Urbach1993, 342–344.) Consider the hypothesis H that the objective probability of the event E is p. According to PP, a rational person's subjective probability of E conditional on H should also be p.

Based on a careful review of the relevant arguments, Howson and Urbach (Reference Howson and Urbach1993, 418) conclude that, within the subjectivist framework, no rationality constraints should be accepted. Nevertheless, they accept PP,Footnote ⁶ arguing that PP is not an additional rationality constraint but a logical consequence of subjectivism (Howson and Urbach Reference Howson and Urbach1993, 344–347).

If Howson and Urbach were right, $C_{2}$ would be false. However, their argument fails (see Section 4), and with it their attempt to rescue frequentism as a theory of relevant probability. Indeed, no Bayesian attempt to rescue frequentism can succeed. This can be shown by making use of $C_{1}$.

The consistency of frequentist-probability statements with determinism allows for the following scenario. Let the deterministic laws generating a sequence of observations be known but let there be uncertainty with respect to future observations because initial conditions are unknown. There are cases where the same information about the initial conditions can be framed in two equivalent ways: either the information is given directly, by specifying the set of initial conditions (deterministic frame), or indirectly, by specifying frequentist probabilities (probabilistic frame).

The deterministic frame yields a paradigmatic example for the application of subjectivism. Subjectivists have always argued that in such cases any prior (or at least any prior from a very large set like the set of all regular priors) is as rational as any other, rejecting principles of insufficient reason or other rationality constraints recommending a single (often, the uniform) prior as the only rational one.

The probabilistic frame, on the other hand, seems to leave room for adopting PP as a rationality constraint. However, it turns out that this is equivalent to enforcing the choice of a single (in the example of Section 4, actually, the uniform) prior in the deterministic frame.

Thus, subjectivists who accept PP as a rationality constraint in the probabilistic frame, but who reject rationality constraints in the deterministic frame, are inconsistent. The inconsistency can be demonstrated by framing the same information in two different ways, leading to inconsistent results concerning the set of rationally admissible priors. The same inconsistency would arise from the adoption of any other rationality constraint overcoming $C_{2}$, because such a constraint operates within the probabilistic frame and restricts the set of rationally admissible priors in comparison with the deterministic frame.

3. Frequentists Toss a Fair Coin

A frequentist model of tosses of a fair coin is an infinite random sequence of 0s and 1s in which the limiting relative frequency of 0 exists and is $\frac{1}{2}$.Footnote ⁷ Randomness can be defined more or less strictly. Generally, randomness requires that an infinite sequence has some property whose absence cannot be discovered by checking any single finite initial segment, but can be discovered recursively, by checking increasingly long initial segments.

Thus, like the limiting-frequency criterion, the randomness criterion refers only to long-run properties of infinite sequences and places no constraints on finite segments of infinite sequences, as stated in propositions LR and IR of Section 2. The stricter the definition of randomness, the more long-run properties an infinite sequence must satisfy to qualify as random, and the more infinite sequences have to be excluded as nonrandom.

As stated in proposition $C_{1}$ of Section 2, infinite random sequences of 0s and 1s with the right limiting relative frequency can be generated by deterministic processes. Consider, for instance, the so-called baker-map dynamics:Footnote ⁸

(1)

The starting point of the dynamical system (1) is some value $Z_{1}\in I\mathrel{\mathop:}= [ 0,\,1) $. The variable $Z_{n}$ stays in I for all n. The variable $Y_{n}$ is either 0 or 1, depending on whether $Z_{n}$ is smaller than $\frac{1}{2}$ or not. From each starting point in I, (1) generates a unique infinite sequences $(Y_{n}) ^{\infty }_{n=1}$ of 0s and 1s.

Thus, we can identify numbers from I with infinite sequences. Some of the infinite sequences generated by the baker-map dynamics do not qualify as a frequentist model of tossing a fair coin. For these sequences, the limiting relative frequency of 0 may not exist or may deviate from $\frac{1}{2}$, or the sequence may violate some definition of randomness. Let $I^{*}\subset I$ be the set of numbers generating sequences that qualify as frequentist models of tossing a fair coin. Subsequently, we restrict the set of starting points of (1) to $I^{*}$ (restricted baker-map dynamics).

It is known that almost all numbers from I (and, thus, almost all sequences) are retained in $I^{*}$, even under the strictest definition of randomness. By definition of $I^{*}$, all sequences generated by the restricted baker-map dynamics have the required long-run properties and qualify as frequentist models of tossing a fair coin. Moreover, each sequence with these qualifications is generated by a unique starting point $Z_{1}\in I^{*}$. Since sequences have been excluded only in view of their long-run properties, no finite initial segment has been ruled out.

4. Enter the Bayesians

It is yet an open question, however, whether a decision maker should adopt $\frac{1}{2}$ as (subjective) probability for 0 in the single case upon the information that the frequentist probability is $\frac{1}{2}$.

Let us apply the reasoning of Howson and Urbach (Reference Howson and Urbach1993, 345) to Eve's problem. The next paragraph partly paraphrases their text.

Suppose Eve is to state her degree of belief in the event $X_{1}=0$, conditional on the information only that the sequence, if extended to infinity, has the stated limiting-frequency properties. Suppose that she were to answer by naming some number $p^{\prime }$ not equal to $\frac{1}{2}$, say, $p^{\prime }=\frac{3}{4}$. Then, according to the definition of degree of belief (Howson and Urbach Reference Howson and Urbach1993, Chapter 5), she believes that there would be no advantage to either side of a bet on the event $X_{1}=0$ at odds 3:1. But that toss was specified only as a member of the sequence with the limiting relative frequency of $\frac{1}{2}$. Hence, as Howson and Urbach would have us conclude, she has implicitly committed herself to the assertion that the fair odds on $X_{n}=0$ for any n, conditional just on the same information that they are members of a sequence with limiting relative frequency of $\frac{1}{2}$, are 3:1.

Given these premises, the argument takes off. If Eve accepts as fair any odds deviating from 1:1 for each single toss, then she would be objectively mistaken. One side of the bet must be more advantageous in the sense that, for an infinite sequence of bets, the accumulated gains from taking this side must go to infinity, while the accumulated gains from the other side must go to minus infinity.

However, the fact that the sequence has limiting relative frequency $\frac{1}{2}$ implies nothing about its beginning. Yet, the conclusion of Howson and Urbach rests on the premise that the probability assignments to the events $X_{n}=0$ in the hypothetical infinite sequence must be identical for all n.

This premise could be justified in one of two ways. One might assume that, objectively, all sequences are equally likely. However, this creates a regress, because we then have to speak of a sequence of experiments where sequences of heads and tails are chosen. On this higher level, the same problem crops up again.

The other justification is Howson and Urbach's, which says that all elements of the sequence must be treated equally because the information about them is the same. But this justification is a version of the oldest principle of insufficient reason: unless there is information to the contrary, assign the same probability to different events. Laplace used this principle to assign the same probability to the different possible outcomes of one experiment. Howson and Urbach use it to assign the same probability to the same outcome in different repetitions of the same experiment. Nevertheless, their arguments against the classical version apply equally to their own version (Howson and Urbach Reference Howson and Urbach1993, Chapter 4).

To this line of criticism, one might object that, actually, Eve is not able to treat different elements of the sequence differently. Contrary to the assumption above, Eve does not know that she bets on $X_{1}$. According to Howson and Urbach, the only information she has is that she bets on some element from a sequence with certain long-run properties. Since she does not know on which element she bets, she cannot treat this one differently from the others.

But this objection is not valid. $X_{1}$ is just the name we give to the element on which Eve actually bets. Therefore, when Eve receives the information that she bets on some element of an infinite sequence in which 0 has frequentist probability $\frac{1}{2}$, she can restrict her considerations to the infinite subsequence beginning with the element she bets on. This infinite subsequence has again the same long-run properties as the original sequence. She can very well have different beliefs about different elements of this subsequence; for instance, she can believe that the second element will turn out to be different from the first.

The conclusion of Howson and Urbach that Eve should choose the same subjective probability for all elements of the infinite sequence is an implicit appeal to a principle of insufficient reason, contradicting their well-founded rejection of such principles.

Thus, Howson and Urbach fail to forge a connection between frequentist probabilities and subjective probabilities. This is not their fault: no subjective Bayesian can forge this connection.

In order to see this, assume that Eve happens to believe that the sequence of observations is produced by the restricted baker-map dynamics, that is, by (1) with starting point $Z_{1}=I^{*}$. All starting points generate random sequences with the required limiting relative frequencies. Therefore, her belief is consistent with the frequentist-probability information she has.

Conditional on the hypothesis that (1) generates the sequence she observes, Eve's information can be framed in two equivalent ways. In the deterministic frame, the information is that the initial condition satisfies $Z_{1}\in I^{*}$. In the probabilistic frame, the information is presented without reference to initial conditions by the statement that the frequentist probability of 0 is $\frac{1}{2}$.

The deterministic frame already commits subjectivists to a position. Predicting the outcome of the restricted baker-map dynamics is a task of the kind for which the subjective Bayesianism advocated by Howson and Urbach has originally been devised. If subjectivism means anything, it means that Eve is allowed to choose almost any subjective prior on the set of possible starting points of a given dynamical system. Howson and Urbach (Reference Howson and Urbach1993, 418) insist that there are no rationality constraints on the choice of prior. Other subjectivists may favor some weak constraint like regularity. In any case, a large set of admissible priors remains.

If Eve chooses the uniform prior as a prior on the set $I^{*}$ of starting points for (1), she implicitly assigns subjective probability $\frac{1}{2}$ to $X_{i}=0$ for all i. However, she is free to choose any other prior on $I^{*}$. By choosing other priors, Eve can assign arbitrary probabilities to the first n observations, for any finite n. Thus, there are no constraints on Eve's subjective probability assignments in the finite case, as stated in proposition $C_{2}$ of Section 2.Footnote ⁹

The probabilistic frame seems to leave room for the Principal Principle (PP) as a rationality constraint. However, since PP requires Eve to assign subjective probability $\frac{1}{2}$ to $X_{i}=0$ for all i, this would be equivalent to enforcing the uniform prior as the only rational choice in the deterministic frame. This should immediately convince subjectivists of all stripes that PP is not acceptable as a rationality constraint.

Moreover, weaker principles connecting frequentist and subjective probabilities should also be rejected. The deterministic frame commits subjectivists to the view that Eve can rationalize any finite betting strategy by some prior. On pain of becoming inconsistent, subjectivists cannot take a different position in the probabilistic frame and accept some principle restricting, if ever so slightly, Eve's choice of betting strategies. After all, whichever frame is used to present the information, it is always the same information.

This completes the argument. Frequentist-probability information puts no restrictions on any finite segments of the infinite sequences; hence, when considering the first, finite part of an infinite sequence (hypothetical or not), Eve is completely free in her beliefs. Should she happen to believe that the sequence is produced by the restricted baker-map dynamics, she is free to speculate about the starting point since all starting points generate the required limiting relative frequencies and are random according to the frequentist definition. Should she happen to believe in indeterminism, it seems that she should have at least as much freedom in forming her beliefs as under determinism.

Therefore, a subjectivist should feel free to ignore frequentist probabilities during his or her lifetime or longer: they are not relevant to any finite number of decisions.Footnote ¹⁰

5. Conclusion

The frequency theory is consistent with determinism because frequentist-probability information is only concerned with the long-run properties of infinite sequences. For the same reason, however, subjective Bayesians should insist that a rational person can adopt any probabilities for finite sequences even in the face of frequentist-probability information.

This implies the rejection of any supplementary principle forging a connection between frequentist and subjective probabilities. Specifically, it implies the rejection of the Principal Principle (PP).

Intuitively, however, PP is right. Beliefs about objective probabilities should constrain decision making. Beliefs about limiting relative frequencies cannot do that. Those who use objective probabilities in decision making, among them some Bayesians and all classical statisticians, must look elsewhere for a justification of their practices. Objective single-case probabilities (propensities) might succeed where limiting relative frequencies fail. However, there might be a price to pay: according to their proponents, most propensity theories are inconsistent with determinism.