2.1. Propositions, Probability, and Measurability

For the purposes of this discussion, I treat the objects of doxastic attitudes as propositions and think of propositions as sets of possibilities.Footnote ² Each proposition is identified with a subset of the set of possibilities for the agent.

I will also assume that the synchronic norms of probabilism have already been established, perhaps by means of the argument that Leitgeb and Pettigrew use, or perhaps independently. Thus, I will assume that the agent’s credences satisfy the axioms of probability theory, and, to emphasize this fact, I will use ‘P’ as the name for the function that represents the credences. To be explicit, this means:

1. 1. Whenever is defined, .

2. 2. .

3. 3. If A and B are disjoint subsets of , then, whenever and are both defined, is defined and equal to .

The condition that these credences are defined turns out to be important for the generalization to the infinite case. It is well-known that in many infinite sets, some natural constraints on probabilities are incompatible with every subset having a probability. Instead, we must assign probabilities only to some collection of the ‘measurable’ subsets of the space. In standard measure theory, it is assumed that is a σ-field. This means:

1. 1. is a member of .

2. 2. If A is in , then (the complement of A in ) is in .

3. 3. If are countably many members of , then their union is in .

By suitable applications of these assumptions, one can show that countable intersections of members of are in too.

Thus, I will not assume that the agent has credences in every proposition, but I will let be the set of propositions for which she does have credences, and I will assume that is a σ-field. Of course, if the agent does have credences in every proposition, then is trivially a σ-field. I suspect that the closure properties of can be weakened somewhat, though this may complicate some of the arguments.

2.2. Updates and Experiments

Not all updates fall in the scope of my argument. I will only consider updates on which remains fixed. Situations in which the agent comes to have credences in new propositions seem very different from the standard examples where an agent just learns that some proposition is true. For instance, two natural cases where would be expected to change are the introduction of a new theory, and the discovery of confusion in one’s old concepts, either one of which might be part of something like a scientific revolution. Bayesians already know (Earman Reference Earman1992) that these cases are difficult ones to account for.

I will follow Greaves and Wallace in considering an update situation as being given in advance by some experiment: “[An experiment is] a situation in which the agent is to receive some new piece of information, from among a set of mutually exclusive and jointly exhaustive alternatives. Mathematically, an experiment is represented by a partition of ” (Greaves and Wallace Reference Greaves and Wallace2006, 611). To say that is a partition of just means that is a collection of propositions such that each member of is in exactly one of the propositions in . Since their is finite, so is any experiment for them. But I will allow to be infinite.

As a simple example, we can imagine that the agent is about to flip a coin and observe how it lands. We can think of as the set , where H is the set of all possibilities in which the coin lands “heads,” and T is the set of all possibilities in which the coin lands “tails.” For a more complicated example, consider the experiment of consulting an oracle to ask how many humans will ever have lived. In this case, it seems reasonable that any sufficiently large integer is an epistemically possible outcome, though we might imagine that there are other possible outcomes as well, such as one in which the oracle says, “I don’t know.” The experiment is represented by the set of distinguishable outcomes for the agent—in this case, rather than one member of being the set of worlds in which there are exactly 10 trillion humans over the history of the universe, the relevant member of is the set of worlds in which the oracle says “10 trillion” in answer to the question. Yet more complicated examples may arise in cases where the agent is about to throw an infinitely thin dart at a dart board—if the agent can distinguish each infinitely small point of the board, then the elements of will correspond to each possible landing site of the dart, but if the agent cannot distinguish these points perfectly, then the elements of will be much more complicated.

As suggested by my examples, the sort of experiment I consider is one in which the agent comes to have credence 1 in exactly one member of and credence 0 in all the rest. Cases in which the update is generated in some other way (e.g., cases driven by conditional probability [van Fraassen Reference van Fraassen1981], cases involving potential memory loss [Arntzenius Reference Arntzenius2003], and cases calling for Jeffrey conditionalization—or the alternative to it that Leitgeb and Pettigrew [Reference Leitgeb and Pettigrew2010b, 252–60.] propose) are beyond the scope of the arguments considered here. However, the notion of conditional probability is traditionally thought to play an important role in these updates, and if the concept of conditional probability can be clarified by my arguments, then they will at least indirectly help us understand these more complicated updates.

I will also assume that is a subset of —each distinguishable outcome of the experiment had some credence for the agent prior to the experiment being performed. In cases where some elements of are not in , the agent has a possibility of coming to know some proposition that she does not already grasp—thus, they are outside the scope of my arguments. Updates involving indexicals like ‘here’ or ‘today’ have this feature, so my arguments do not directly apply to “Sleeping Beauty” or related cases (Meacham Reference Meacham2008; Titelbaum Reference Titelbaum2008; Bradley Reference Bradley2012).

2.3. Plans and Availability

I will consider an update as proceeding according to a set of “plans” for an experiment.Footnote ³ For a given experiment , and for each proposition A in , I will represent the agent’s plan for updating her credence in A after performing as a function that gives a real number in each state. For any state s in , is the credence that the agent plans to have in A after learning the outcome of , if s is the actual state. When is understood from context, I will sometimes refer to just .

This use of “plans” distinguishes my framework slightly from those of Leitgeb and Pettigrew, and Greaves and Wallace. Leitgeb and Pettigrew think of an update as given by a single proposition E that the agent becomes certain of. In the case where is finite and all its members have positive prior credence, it will turn out that the difference is inessential. However, when some members of have prior credence 0 (and in many cases, such as ones where the experiment consists of learning the precise real value of some unknown physical parameter, it may be the case that all members of have prior credence 0), the partition will play an essential role in the argument. And once we have a partition rather than a single proposition, something like this notion of “plan” is essential.

Greaves and Wallace talk of “acts” whereas I talk of “plans.” The main difference is that while their acts specify a complete probability distribution for the agent in each state, my plans only specify the agent’s credence for a single proposition. This primarily helps simplify some of the discussion, but it is also connected to the distinction between global and local inaccuracy measures, to be discussed in section 2.4.Footnote ⁴

Of course, given the notion of updates and experiments I am working with, not just any such function will correspond to a possible plan. Only a subset of these functions will be said to be “available.” Since the experiment represents the information the agent comes to know during the course of the update, an agent’s plans must make the same response in states where the outcome of the experiment is the same.

A plan is said to be available (or “-available,” if I need to make explicit which partition is relevant) if and only if it meets the following two conditions:

1. 1. If and , then .

2. 2. If x is any rational number, then , where is the set of states s for which .

A plan that did not satisfy the first constraint would require the agent’s credence function to depend on information that she does not have, even once the experiment is performed. The terms and are both compatible with all the information that she has so far (which is what it means to be in ), and since what she learns on completing is just which member of is actual, she does not learn anything to distinguish them. Thus, she cannot plan to act differently in those two states.

The second constraint is a bit more subtle. If is a finite or countable partition, then it is not a further requirement but in fact follows from the first one, together with the fact that is a σ-algebra. So it is only a further assumption if is uncountably infinite. In planning whether to have credence greater than x, the agent is responding to whether the actual world is in or not, and it seems that this responsiveness should require that be a proposition she can grasp. (Readers who are familiar with the language of measure theory will recognize this second constraint as the requirement that be measurable with respect to the agent’s credence function, which is essential for the notion of expected value to make sense.)

2.4. Inaccuracy, Global and Local

The arguments under consideration are all based on a function I that measures ‘inaccuracy’ of a belief state. The one I will use is what Leitgeb and Pettigrew call a ‘local inaccuracy measure’, which is a function such that is a real number that measures how epistemically bad it is to have credence x in proposition A in state s. Greaves and Wallace instead describe their function as an ‘epistemic utility function’, but in the terminology of Leitgeb and Pettigrew, it behaves like a ‘global inaccuracy measure’. This is a function U such that is a real number that measures how good or bad the overall credence function p is in state s.

Leitgeb and Pettigrew argue that agents should have both local and global inaccuracy measures and that these should cohere in a particular way. As a result, they claim that the local inaccuracy function should be given by the Brier score, which is defined by if A is true in s and if A is false in s. They also argue that the global inaccuracy function should be given by the sum of the local inaccuracies over all propositions. Their arguments are interesting, but they rely on some controversial assumptions about the “geometry of reason” (Leitgeb and Pettigrew Reference Leitgeb and Pettigrew2010a, 210; see also their theorems 3, 4, and 5 on 222–29). In addition, these arguments cannot work when there are infinitely many propositions to be considered—their version of global inaccuracy will normally be infinite and thus cannot relate to local inaccuracy in any interesting way.Footnote ⁵

Greaves and Wallace make no claims about local inaccuracy measures and very few claims about the global inaccuracy measure that they use. This gives them some greater generality but also requires that they consider an update in terms of “acts” that specify posterior credences for all propositions, rather than my “plans” that specify posterior credence for only a single proposition at a time.

I will instead work just with local inaccuracy measures. I will make five assumptions about the local inaccuracy function . (These assumptions are weaker than the ones that Leitgeb and Pettigrew make and have parallels in other discussions of inaccuracy, such as Joyce Reference Joyce1998.)

Bounded Continuity. Third, I assume that for any positive ε and each A there is a set of finitely many values such that, for every ,

and

(Given Monotonicity, this is equivalent to the functions being bounded and continuous on .)

Figure 1.

These five assumptions are all satisfied by the Brier score, so they follow from the assumptions made by Leitgeb and Pettigrew. They also follow from Joyce’s assumptions on accuracy and are satisfied by the other scoring rules discussed on page 275 of Joyce (Reference Joyce, Huber and Schmidt-Petri2009). Greaves and Wallace assume only the equivalent of Immodesty for their global inaccuracy measure. But since this is the most arbitrary-seeming assumption, my argument is no worse off than theirs.

2.5. Expected Inaccuracy

Before giving the proofs, I need to explain a bit more about ‘expected inaccuracy’ and show why this notion makes sense. If V is a random variable (i.e., a function that takes on a real value in each state s in ), then the ‘expected value of V’, notated as , represents something like the agent’s best estimate of V. It depends on the values for V in each possible state and the agent’s credence function P. Although may be distinct from every possible value of V (just as the average family may have 2.3 children even though no particular family has exactly that many), it traditionally plays a role in guiding one’s actions in light of one’s uncertainty about V.

We can stipulate a definition of the expected value of V with respect to P, as Leitgeb and Pettigrew do. However, they consider at least two different stipulations (, or ), which give distinct results for some infinite cases, and both seem clearly wrong for others.Footnote ⁶ And there are further worries that arise in taking a sum of infinitely many terms, especially if the series is uncountable.

Thus, instead of stipulatively defining expectation as Leitgeb and Pettigrew do, I will adopt the axiomatic approach of Whittle (Reference Whittle2000). For random variables , Whittle’s axioms state:

1. 1. If for all s, then .

2. 2. If c is a constant, then .

3. 3. .

4. 4. For an event A, let be the random variable with if A is true in s and if A is false in s. Then .Footnote ⁷

5. 5. If for every s the sequence is monotonically increasing and converges to , then the sequence is monotonically increasing and converges to .

Axiom 1 seems essential to the concept of an expected value. A random variable cannot have a negative expected value unless it is at least possible for it to take on a negative value. Axiom 4 gives the only direct connection between probability and expected value. Axioms 2, 3, and 5 provide linearity conditions on expected value. Axiom 3 is often a surprise to students when they learn that it holds regardless of whether and are independent or dependent. However, once one has convinced oneself that it holds, it is plausible that it is in fact essential to the notion of expected value and not merely a mathematical consequence of a formal definition.

If V is a measurable random variable that always takes either the value v or the value 0, then axioms 2 and 4 guarantee that . If V is a measurable random variable that takes on only finitely many possible values, then it is the sum of finitely many random variables of the previous sort, and thus, applying axiom 3, we see that .

If V is a measurable random variable, and it is bounded (there are a and b with for all s), then this previous result together with axiom 5 will be sufficient to uniquely determine . This is because we can define a sequence of random variables that are monotonically increasing and converge to V, each one of which is measurable and takes on only finitely many possible values. In particular, we define , and we define whenever . (See fig. 2.)

Figure 2.

The result of the previous paragraph suffices to define each , and axiom 5 guarantees that is the limit of these values. (This is in fact equivalent to the standard measure-theoretic definition of the expected value of a bounded measurable random variable as a Lebesgue integral.)

If we let be some available plan for updating one’s credence in A for some experiment , and if is the inaccuracy , then we can see that V is measurable (because is measurable, so is measurable, and I satisfies Extensionality and Monotonicity, so it is measurable as well) and bounded (because I satisfies Bounded Continuity). Thus, Whittle’s axioms for expectation suffice to define the expected inaccuracy of any plan of the sort I consider.

Without Extensionality, inaccuracy might depend on some proposition for which the agent does not have a credence, so it would not have an expectation. Without Monotonicity and Bounded Continuity, inaccuracy could be unbounded in a way that results in its expected value being infinite or undefined. Thus, some version of these first three assumptions is surely needed for expected inaccuracy to be the sort of thing one could try to minimize.

The use of expected inaccuracy distinguishes the arguments of Leitgeb and Pettigrew, and that of Greaves and Wallace, from some other accuracy-based arguments. For instance, Lindley (Reference Lindley1982) and Joyce (Reference Joyce1998) both give arguments for probabilism (Lindley argues for the ratio analysis of conditional probability as well) based merely on the requirement that no other credence function dominate the one under consideration (one credence function “dominates” another if and only if the first has lower inaccuracy in every state). Since one credence function can have lower expected inaccuracy than another without being dominated, the Joyce and Lindley results are correspondingly stronger. All of these arguments, other than that of Greaves and Wallace, assume a strong connection between global and local accuracy.

My argument (like that of Leitgeb and Pettigrew) makes the stronger assumptions of each, that utility should be defined by a local inaccuracy measure and that agents should rule out all plans with higher expected inaccuracy rather than merely plans that are dominated. If there is some problem in principle with local inaccuracy measures or expected inaccuracy, then this would give a reason to reject my conclusions while accepting the conclusions of some of the others. However, my arguments apply even when is infinite, and I give results leading to versions of conglomerability and reflection, in addition to the ratio formula for conditional probability. (As an anonymous referee has pointed out, the authors that give dominance arguments rather than expectation arguments are trying to establish probabilism as well as conditionalization. If probabilism is not already taken for granted, then the use of expectation is clearly problematic. But once probabilism is granted, as I have assumed here, the use of expectation rather than dominance may not be a real weakness.)

2.6. The Theorems

The following results are proved in the appendix.

This theorem is the same as theorem 3 in Leitgeb and Pettigrew (Reference Leitgeb and Pettigrew2010b, 250), except for my weaker assumptions on I and the measurability assumptions needed for the infinite case. This is also the same as corollary 1 that Greaves and Wallace state on page 625, except that they only assume Immodesty, while I use a local inaccuracy measure and the measurability assumptions for infinite sets.

This theorem basically says that no optimal update plan guarantees an increase in credence, or guarantees a decrease in credence. For every experiment that could increase one’s credence in A by some positive ε, there is always some other possible outcome of the experiment that would fail to do so.

In many cases, the content of this theorem can be better understood by looking at a more specific consequence:

That is, the expected result of an update plan for A should equal one’s prior credence in A.

All of these theorems tell us what plans that minimize expected inaccuracy must be like, if they exist. Fortunately, such plans do exist. (The space of possible plans is the set of functions from to , which is compact, by Tychonov’s theorem. Expected inaccuracy is a continuous function on the space of plans and must therefore achieve its minimum value.)

The extra condition in the statement of the Expectation Theorem is a bit subtle. In particular, since I have a separate plan for each proposition, I have not assumed that the plans will combine to yield a coherent posterior probability function. And in fact, if is a countable partition of propositions whose credences fail to sum to 1, then it is known that the conclusion of the Expectation Theorem must be violated, so there can be no set of plans that cohere probabilistically (Hill and Lane Reference Hill and Lane1985, 369). However, if the unconditional credences satisfy countable additivity (as orthodox Bayesianism assumes), then the Radon-Nikodym theorem of real analysis guarantees the existence of coherent update plans that satisfy the conclusions of all three theorems. It remains to be seen if such plans must always minimize expected inaccuracy, but it seems plausible that they will.

When is finite, the Interval Theorem and Expectation Theorem are trivial consequences of the Ratio Theorem. It is only in the infinite case that these provide additional content. Thus, the normative claims about updating and conditional credence that I discuss in the next section all apply to the finite case as a consequence of the arguments by Leitgeb and Pettigrew or Greaves and Wallace. My contribution is to show that they hold in the infinite case as well.

3.1. Accuracy Only?

In much of the literature on accuracy, it is assumed that accuracy is the only normatively significant feature of a credence function. Leitgeb and Pettigrew (Reference Leitgeb and Pettigrew2010b, 244) defend this by arguing that “the ultimate desideratum for a belief function is that it be close to the truth” and that any other virtue of a belief function can be sacrificed if it conflicts with accuracy. In forthcoming work, Pettigrew suggests further that various other desiderata, like being appropriately responsive to one’s evidence, may also be derived from accuracy, so that these other values cannot conflict with accuracy. Similarly, Joyce (Reference Joyce, Huber and Schmidt-Petri2009, 275) suggests (at least tentatively) that epistemic value reflects considerations of accuracy alone, though he also suggests it may be merely a “useful fiction” (266). (Greaves and Wallace skip accuracy altogether and just talk about ‘epistemic utility’.)

I will follow these authors in assuming that the inaccuracy measures include everything that is relevant. If there are other normatively significant values that could potentially overrule accuracy, then our results will all have to be interpreted quite differently (Easwaran and Fitelson Reference Easwaran and Fitelson2012). But whatever support their results give to probabilism and conditionalization, my results give the same support to some additional norms for infinitary cases. The use of norms based only on accuracy is what I will call “the accuracy framework.” As stated in the introduction, the point of my argument is just to illustrate the power of the accuracy framework in cases with infinitely many possibilities. It is a further question whether this power is a strength or a weakness.

3.2. Synchronic Norms on Conditional Probability

First I will see what my theorems tell us about the doxastic state of an agent at a time. I will start by assuming the following synchronic norm on plans (here, and in everything that follows, P will represent the agent’s current credences, before actually performing the experiment):

This then gives us norms on plans from our theorems. From the Ratio Theorem we get:

From the Interval Theorem we get:

From the Expectation Theorem we get:

Of course, an agent’s plans for updating are not usually the sorts of things we talk about in discussing the synchronic status of an agent’s credences. However, one might think that these plans have a normative connection to her current conditional credences. I propose:

Of course, this norm is trivial if one thinks that conditional credences just are one’s plans for updating. For reasons discussed in Hájek (Reference Hájek2003) and Eriksson and Hájek (Reference Eriksson and Hájek2007), I suspect that update plans and conditional credence probably are not actually the same state. However, it seems plausible that there is a normative connection between the two.

Given both these norms, we can get the following additional conclusions. The Ratio Theorem gives:

The Interval Theorem gives:

The Expectation Theorem gives:

Ratio Analysis is of course the standard account of conditional probability, and the other two are the principles known by those names in the literature (Dubins Reference Dubins1975). Philosophers and statisticians have given various arguments against Conglomerability and Disintegrability (Hill Reference Hill1980; Hill and Lane Reference Hill and Lane1985; Kadane et al. Reference Kadane, Schervish, Seidenfeld, Goel and Zellner1986; Arntzenius et al. Reference Arntzenius, Elga and Hawthorne2004). If these challenges are taken to be decisive, then this raises problems for the accuracy framework. Defenders of the accuracy framework will have to use some of the defenses suggested by Easwaran (Reference Easwaran2008) or suggested by Dickey, Fraser, or Lindley in their responses included in Hill (Reference Hill1980).

3.3. Diachronic Norms on Updates

Leitgeb and Pettigrew, and Greaves and Wallace, claim to get more than just synchronic norms on credences. They both claim that over the course of an update, an agent should obey the following norm:

A natural diachronic norm that gives Conditionalization as a consequence of the Ratio Theorem would be:

In light of my Interval Theorem and Expectation Theorem, we see that any update satisfying my conditions should also satisfy:

Principles like this were discussed and attacked by van Fraassen (Reference van Fraassen1984), and many other counterexamples have been proposed since then. But to the extent that these counterexamples involve updates that violate the assumptions I made in section 2.2, they are not challenges to my normative claims. Any counterexamples that involve only finite partitions are just as much challenges to Conditionalization as to the various reflection principles.

But regardless of the status of the counterexamples, I am somewhat suspicious of any such diachronic norm. As Michael Titelbaum pointed out in comments on Greaves’s presentation of Greaves and Wallace (Reference Greaves and Wallace2006) at the 2006 Pacific meeting of the American Philosophical Association, if an agent just arbitrarily updates in a way that violates her prior plan, and we use the agent’s new credence function to calculate the expected accuracy rather than the old function, then Immodesty guarantees that any update to a probabilistically coherent set of credences will end up minimizing expected inaccuracy. (Titelbaum recommended instead a synchronic interpretation of the significance of their theorem, like what I considered in sec. 3.2.) Thus, to get this conclusion, we need a real diachronic norm on credences.

Giving a diachronic norm on credence is a notoriously difficult problem. Despite the fact that a variety of arguments formally reach diachronic conclusions, Christensen (Reference Christensen1991, 246) argues that “without some independent reason for thinking that an agent’s present beliefs must cohere with her future beliefs,” there can be no more support for a diachronic norm (like conditionalization) than for an interpersonal norm (e.g., the requirement that one’s credences must match those of one’s spouse).

Arguments based on decision theory and expected values are no better at providing reasons for intertemporal coherence. Various authors have tried to give arguments that the very fact that an agent has made a plan gives her a reason to stick with it when the appropriate time comes (McClennen Reference McClennen1990; Gauthier Reference Gauthier1994; Hinchman Reference Hinchman2003). However, there are some decision situations that are apparent counterexamples, where it seems an agent could do better by abandoning her plans (Kavka Reference Kavka1983).

Thus, van Fraassen (Reference van Fraassen1995) moved from attacking General Reflection to defending principles more like Plan Reflection, saying that it is only one’s synchronic commitments that are susceptible to these normative requirements, rather than one’s actual diachronic updates.

Another perspective on reflection principles is provided by Skyrms (Reference Skyrms1990), who suggests that they should be taken as a way to classify certain updates as instances of “learning,” even though they do not hold for all updates. (See the discussion on pages 97 and 99 of Skyrms [1990] of “Condition M,” which is equivalent to General Reflection 2.) It is interesting that Myrvold (Reference Myrvold2012) has shown that, for an accuracy measure that satisfies Immodesty, any update that satisfies Condition M must decrease expected inaccuracy. My results can be seen as a sort of converse—an update that satisfies my conditions and minimizes expected inaccuracy must satisfy Condition M. The main difference is that they use Condition M to characterize the updates in question, while I show that it holds for updates that satisfy the conditions given in section 2.2.

3.4. Upshot

Thus, using the accuracy framework, we get some support for diachronic reflection principles and synchronic principles of Conglomerability and Disintegrability, in addition to the support for conditionalization given by Greaves and Wallace, and Leitgeb and Pettigrew. Although there are worries about the diachronic norms, the synchronic norms already provide interesting constraints on conditional probability, which may still be of use in understanding diachronic updates. I have shown that these results hold even without the idealizing assumption that all partitions are finite (in which case all the results follow from Ratio Analysis and Conditionalization).