1. Introductory Overview

In the Bayesian approach to rational deliberation (e.g., Howson and Urbach Reference Howson and Urbach1993; Jaynes Reference Jaynes2003; Joyce Reference Joyce, Gabbay, Hartmann and Woods2009), an agent indicates how strongly she believes a certain statement by giving that statement a credence between 0 and 1. The closer to 1 this credence is, the more strongly the agent considers the statement to be true. In this approach, credences behave like probabilities and are updated when new information becomes available. Consider then the following story (van Fraassen Reference van Fraassen1981, 376–77):

How should Judy Benjamin update her credences, in particular, her credence that she has landed in the Blue Army’s area?

In classical Bayesianism, the only type of new information for which updating is defined is that which makes the agent become certain of some proposition, say E. If Cr₀ indicates the agent’s prior credence, the credence before the new information became available, and Cr her posterior credence—that is, the credence obtained by updating her prior credence—then

This form of updating is also known as Bayesian Conditionalization. It was extended by Jeffrey (Reference Jeffrey1983) to include new information that would make the agent assign precise credences to the members of a partition, a set of statements of which not more than one can be true and of which exactly one must be true. If we indicate this partition by {B_i}, where B_i is an arbitrary member of the partition, then, according to Jeffrey,

in which {Cr(B_i)} is the new information. This form of updating is often referred to as Probability Kinematics and, formally, includes Bayesian Conditionalization. The accompanying information, the set {Cr(B_i)}, will be referred to as Jeffrey information.

The information that Judy Benjamin receives, however, does not seem to have either of these forms. It is not some proposition that she is made aware of, nor is it a posterior credence function on the members of a partition. Instead, what she is told is the posterior value of a specific conditional credence. No generally accepted rule for updating credences upon the acquisition of that type of information is available. One that has been vigorously defended by some (e.g., Shore and Johnson Reference Shore and Johnson1980; Williams Reference Williams1980) is minimization of the relative entropy:

subject to one or more constraints representing the new information. The set {L_i} is the coarsest relevant partition, and the constraint in Judy Benjamin’s case is the value of a posterior conditional credence.

Uffink (Reference Uffink1995), however, has shown that the arguments in favor of minimizing the relative entropy are not convincing, and van Fraassen (Reference van Fraassen1981), with his Judy Benjamin story, has shown that minimizing the relative entropy can, in fact, lead to counter-intuitive if not outright implausible results. Let ‘B’ indicate ‘I am in the Blue Army’s area’, ‘H’ ‘I am in a headquarters area’, and ‘S’ ‘I am in a second company area’. Finally, let ‘¬B’ stand for ‘not B’. Judy’s credences before she received the radio message were Cr₀(B) = 0.5 and Cr₀(H&¬B) = Cr₀(S&¬B) = 0.25. The new information amounts to Cr(H|¬B) = 0.75. What, for her, is now Cr(B)? Minimizing the relative entropy gives Cr(B) = 0.533, but that is rather counter-intuitive, for why would information about where in the Red Army’s area she has landed, assuming that she has landed in that area, give information about whether she has landed in that army’s area? And, assuming that it does give some information, why would it make her increase her credence in not having landed in that area? In fact, minimizing the relative entropy will make her raise her credence for having landed in the Blue Army’s area, no matter what value of Cr(H|¬B) she is given. That result is implausible and led van Fraassen to comment: “It is hard not to speculate that the dangerous implications of being in the enemy’s headquarters area, are causing Judy Benjamin to indulge in wishful thinking” (379).

Van Fraassen, Hughes, and Hartman (Reference van Fraassen, Hughes and Hartman1986) proposed several alternatives to updating by minimizing the relative entropy and demonstrated that all of them have the same counter-intuitive consequence as minimization of the relative entropy and that none of these alternatives is clearly preferable over the others. Grove and Halpern (Reference Grove, Halpern, Geiger and Shenoy1997) suggested that the problem be embedded in a larger space in which one can conditionalize on the conditional credence. Even though they do obtain the plausible answer that Judy’s credence in having landed in the Red Army region does not change, this approach still has serious problems. It is unclear whether it can easily be applied to more complex Judy Benjamin–like problems, and it introduces the new problem of having to choose a prior credence function in the larger space.

Douven and Romeijn (Reference Douven and Romeijn2011) proposed a minimization procedure that produces the intuitively correct answer in the Judy Benjamin case but which has severe drawbacks in other situations. It corresponds, in fact, to Adams conditioning (Bradley Reference Bradley2005) in which the credence in the antecedent is assumed not to change. This may be a correct assumption in some situations but is clearly not tenable in others. Furthermore, Douven and Romeijn’s arguments in favor of Adams conditioning, even in the context of the original Judy Benjamin problem, are not convincing. It seems plausible that, in the Judy Benjamin story as told by van Fraassen, Judy has little reason to alter her credence in being in the Blue Army’s area because all she learns is where she is likely to be if she is in the Red Army’s area. But in that version her prior credence in being in any of the four possible regions is the same; she is equally convinced that she will land in a headquarters region as that she will land in a second company region. What would happen if her prior beliefs were different?

Consider, for example, the scenario in which Judy’s prior credence in having landed in some headquarters region is very low. For example, the training intentionally included a low and known probability of being dropped off in a headquarters region. When she then learns that, if she has landed in the Red Army’s area, she is most likely to find herself in that army’s headquarters region, it seems more plausible for her to lower her credence for having landed in that army’s area. Keeping it the same, after all, would imply that it is more likely than not that she has landed in the headquarters region of some army, and that consequence is contradicted by the training parameters. Contra Adams conditioning, therefore, her credence in having landed in the Red Army’s area may very well change, depending on her prior credences and the new information she receives. The second scenario (low probability of landing in a headquarters region) leads to a lowering of her credence that she has landed in the Red Army’s area, which is exactly what minimizing the relative entropy predicts.

But it is not just changing prior credences that can lead us away from Adams conditioning or minimizing the relative entropy. It is easy to construct stories that are structurally similar to that of Judy Benjamin and that have identical prior credences but in which the credence in the antecedent clearly changes upon receiving information that is analogous to that which Judy receives. Consider the following story:

I think it is uncontroversial that Harry’s credence that Tom will accept the job offer increases when he learns from Sue that Tom might have to move west if he did accept the offer, for moving west is something he was thinking of doing anyhow. So Adams conditioning will not do. But minimizing the relative entropy will not do either because Harry’s credence in the antecedent, that Tom will accept the job offer, increases rather than decreases as minimizing relative entropy would predict. Not that the latter update technique is always wrong. If Sue had told Harry that the job offer comes with a 3-year commitment in New Jersey, Harry’s credence that Tom will accept the offer would have gone down, which is in agreement with minimization of the relative entropy.

The job story and the Judy Benjamin story are structurally the same because we can map “being in the Red Army’s” territory on “accepting the job offer” and “being in a headquarters area” on “moving west,” which will maintain all the (conditional) credences. They are not the same in all respects, however, and the question may arise whether the difference in the change in the credence of the antecedent can be explained by these nonstructural differences. For one thing, the Judy Benjamin story is indexical: Judy does not know where she is. That difference cannot explain the difference in doxastic behavior, however, because the story could have been told from the viewpoint of an observer in the Blue Army’s area who is able to listen to the radio communications between Judy and the duty officer. His credences concerning Judy’s location are not indexical, but the same analysis can be given for his posterior credence that she is in the Blue Army’s area as for Judy’s credence that she is in that area, with the same implausible result if minimization of the relative entropy is used. A more important difference is that Judy’s posterior credence concerns where she is, which is an already established fact: she is where she is, but she does not know where that is. In the job offer story, however, the relevant posterior credence concerns whether Harry will accept the job offer and when he will move west, the answers to which lie in the future and depend on Harry’s desires. Even though this difference is real, the simplistic rule to use Adams conditioning in the first case and something else in the second case cannot be valid because Adams conditioning is wrong in the Judy Benjamin case after a minor change in her prior credences, while Tom’s posterior credence that Harry will accept the job offer can go both up and down, depending on the details of the job offer. It does not seem from these two examples and the minor variations in the two examples, therefore, that the choice of which mechanism to use to determine posterior credences is tied in any simple way to the details of the story.

The Judy Benjamin story by itself has, of course, only limited value, but it can be generalized to a new type of information that is not of the Bayesian type (one that is suitable for Bayesian Conditionalization) or of the Jeffrey type (suitable for Probability Kinematics). I refer to this new type of information as Judy Benjamin information. It consists of a set {Cr(B_i|A)} of conditional credences, with {B_i} a partition and A some statement for which Cr₀(A) does not vanish. The essential difference between Judy Benjamin type of information and Jeffrey type of information is that the former requires the values of conditional credences pertaining to some partition rather than absolute credences. The central lesson coming out of the Judy Benjamin example is that minimizing the relative entropy is not likely to be the correct way of updating credences in the general case involving Judy Benjamin type of information. This point has been confirmed subsequently by various other authors (most recently by Douven and Romeijn Reference Douven and Romeijn2011).

I should point out, however, that van Fraassen (Reference van Fraassen1989, chap. 13.6) does not seem to see it that way. In his opinion, maximizing the entropy or minimizing the relative entropy is generally valid, at least when the new information consists of expectation values of random variables, which class includes that of the Judy Benjamin variety. That that update method gives strange results in the Judy Benjamin case is, if I understand him correctly, not sufficient reason for him to abandon it. He does allow, however, for a nonunique answer to the Judy Benjamin problem. Moreover, the result that Judy’s credence of being in the Red Army’s region decreases has been argued to be correct by Lukits (Reference Lukits2014).

Minimizing relative entropies is not appropriate in the original Judy Benjamin story, and using some other update mechanism, such as Adams conditioning, is inappropriate when some of the details of the Judy Benjamin story are changed. A single update mechanism when Judy Benjamin type of information becomes available will, therefore, not do. But, Adams conditioning seems, intuitively, to be the correct way of updating credences in the original Judy Benjamin story, and updating by minimizing the relative entropy undoubtedly gives the intuitively correct results in some other scenarios, showing that both seem to be appropriate in at least some scenarios. What seems to be more promising, therefore, is a function that maps each specific scenario with Judy Benjamin type of information to a specific update mechanism. In other words, what might be required is a family of update mechanisms and a function that maps the set of Judy Benjamin type of scenarios into this family of update mechanisms.

I present a plausible family of update mechanisms in section 2, but finding an associated function that maps Judy Benjamin–type problems into this family is less straightforward. In fact, I argue in section 3 that no such function is likely to be found, at least when we demand that it maps each specific problem onto a single update mechanism. I argue, instead, that the best we can do is to have such a function map Judy Benjamin–type problems onto subsets of update mechanisms—even, in some cases, onto the whole family of mechanisms. We may consider a subset having more than one member a consequence of our limited semantic sophistication when confronted with Judy Benjamin–type problems, but, as I further argue, it may also be an inevitable consequence of the nature of such problems in the sense that no increase in semantic sophistication will reduce the subsets to singletons—in other words, that Judy Benjamin–type problems may lead by their very nature to indeterminate posteriors.

This conclusion has obvious relevance for the general problem of how to update credences when new information, of any type, becomes available. That is, it addresses the core of Bayesianism, the idea that the strengths of our beliefs can be modeled by probability-like credences and that the effect new information has on those credences is a determinate update from one credence function to another. The argument alluded to in the previous paragraph indicates that, unlike the severely constrained case of Jeffrey type of information, more general forms of new information lead to posterior credence functions that are, practically or constitutionally, indeterminate. I return to this point in section 4. It answers more fully van Fraassen’s original concern: whether minimizing relative entropies is always the correct way of updating credences upon the acquisition of new information, and, if not, what should take its place. Van Fraassen had already indicated that the answer to the first question is negative (but not in his later work [van Fraassen Reference van Fraassen1989]; see previous comments). The conclusion of this article is that no other single update mechanism will do either and that, for general forms of new information, including Judy Benjamin type of information, updating credences may produce indeterminate posteriors.

2. A Universal Family of Update Mechanisms

In this section, I present a family of possible mechanisms for updating credences when confronted with Judy Benjamin type of information. I focus on the posterior credence of the (negation of the) conditioning statement.

Let {B_i} be a partition, A such that Cr(A) > 0, and {Cr(B_i|A)} the set of newly acquired posterior credences of the members of the partition. Unless otherwise mentioned, the partition has n members, and I assume throughout this article that the prior credences of the members of the partition are nonzero. The posterior conditional credences are nonnegative and sum to 1. How {Cr(B_i|A)} is obtained is not part of the update process. It is presumably conveyed to the agent by means of conditional statements and may involve odds ratios, conditional credences, and the like. Interpreting conditional statements is notoriously difficult, and I assume that the information, in whatever form it was given originally, has been translated correctly to the set {Cr(B_i|A)}. New information of the Judy Benjamin type is similar to Jeffrey information, be it that the former is based on the partition {¬A, B₁A, … , B_nA} rather than the partition {B_i}. The new credences of the members of the former partition are given by Cr(¬A), and Cr(B_iA) = Cr(B_i|A)(1 − Cr(¬A)), with Cr(¬A) unknown.

A powerful way of determining posterior credences is by minimizing some functional of the prior and posterior credences under appropriate constraints, with this functional having the property that it is minimal when prior and posterior credences are the same. Shore and Johnson (Reference Shore and Johnson1980) have studied plausible properties that any such functional should have, and Uffink (Reference Uffink1995) showed that their axioms are met by the members of the family of α-divergences U_α(Cr, Cr₀) (also referred to as Rényi relative entropies), defined as

in which the sum is taken over the members of the coarsest relevant partition ({L_i}), and

If Cr₀(L_i) = 0 for some i, U_α is infinite (and, therefore, not a minimum) unless (a) Cr(L_i) vanishes as well and (b) r(L_i) is assumed to be finite. The family of α-divergences will be indicated by F. Clearly U₀(Cr, Cr₀) = 0 and does not correspond to a usefully minimizable functional. We will see that it can be taken to correspond to Adams conditioning. The case of α = 1 is more interesting. It is not defined directly by equation (4), but, by taking the limit of α going to 1, we find the relative entropy (eq. [3]).

Divergence U_α(Cr, Cr₀) has a number of attractive features, in addition to meeting the Shore and Johnson axioms. First, it includes the relative entropy, as we already saw. Second, it is nonnegative and equal to zero if and only if Cr(B_i) = Cr₀(B_i) for all members of the partition because x^1−α is a concave function of x for 0 < α < 1 and a convex one otherwise. Third, U_α(Cr, Cr₀) reproduces Probability Kinematics when Jeffrey type of information becomes available. This crucial property can be demonstrated easily and completely generally. Consider an arbitrary statement E for which we want to determined the posterior credence: E gives rise to the partition {EB₁, … , EB_n, ¬EB_i, … , ¬EB_n}. We proceed by minimizing U_α(Cr, Cr₀) for this partition with the Jeffrey constraint

and known {Cr(B_i)}. Equation (4) becomes

Note that r(EB_i) = Cr(B_i)Cr(E|B_i)/Cr₀(EB_i) and r(¬EB_i) = Cr(B_i)(1 − Cr(E|B_i))/Cr₀(¬EB_i) and that the members of {Cr(E|B_i)} can be varied independently in [0, 1]. Since both Cr₀ and {Cr(B_i)} are known and fixed, minimizing U_α is accomplished by minimizing it with respect to each Cr(E|B_i). This leads to r(EB_i) = r(¬EB_i), or Cr(E|B_i) = Cr₀(E|B_i). The matrix of second derivatives is easily shown to be positive and definite. The equality between Cr(E|B_i) and Cr₀(E|B_i) is called rigidity and is a consequence of the particular form of U_α. Equation (2) follows immediately because Jeffrey Conditionalization can be proven when rigidity is assumed (Diaconis and Zabell Reference Diaconis and Zabell1982).

With a Judy Benjamin constraint, the appropriate partition is {¬A, B₁A, … , B_nA}, and {Cr(B_i|A)} is given. I restrict the analysis to the case that Cr₀(B_iA) > 0 for all i. To simplify subsequent expressions, I abbreviate Cr₀(B_i|A) by p_i and Cr(B_i|A) by q_i: q_i is known and positive for each i, and ∑q_i = 1. The Judy Benjamin constraint becomes Cr(B_iA) = q_i(1 − Cr(¬A)) for i = 1, … , n, with Cr(¬A) unknown. We proceed as follows. We first determine Cr(¬A) by minimizing U_α with the underlying partition {¬A, B₁A, … , B_nA}. Equation (4) becomes

Minimizing with respect to Cr(¬A) leads to the equation r(¬A) = r(A)D with

The final result is

Equation (10) will be referred to as the Judy Benjamin equation. Clearly, when α = 0, D = 1, and Cr(¬A) = Cr₀(¬A). As we saw before, U₀ is constant and, therefore, not suitable for minimization, but we can replace that member of the family of α-divergences with Adams conditioning. When α = 1, we need to take the limit of α going to 1, and find .

Once Cr(¬A) has been determined, the credences of all the members of the partition are known, but then we are back at a standard Jeffrey problem, for which the solution is

for the calculated value of Cr(¬A). Note the extreme simplicity of the result: when determining a posterior distribution for Judy Benjamin type of information by minimizing an α-divergence, we find Jeffrey Conditionalization again. The input that drives the update is {Cr(B_i|A)}, and Cr(¬A) is obtained from the Judy Benjamin equation. All α-divergences produce a Jeffrey Conditionalization result; they differ only in their values of Cr(¬A).

In general, the Judy Benjamin equation requires a simple calculation once D is known. The question now is how Cr(¬A) varies when α is varied. First, note that D is nonnegative and equal to 1 when α = 0. Second, Cr(¬A) is a monotonically increasing function of D, and, therefore, we need to determine only how D varies with α. Third, D is a continuous monotonically increasing function of α. For α > 1, this follows immediately from the Hölder inequality. For α < 1, define C = 1/D and write . Finally, the derivative with respect to α at α = 1 exists and is positive. Consequently, Cr(¬A) is a monotonically increasing function of α as well: it is less than Cr₀(¬A) when α is negative and larger than Cr₀(¬A) when α is positive.

Because of the monotonicity of Cr(¬A) as a function of α, the possible values of Cr(¬A) are bounded by their values for α = −∞ and α = ∞. In equation (9), the ratios q_i/p_i can take all values in (0, ∞). Let s be the index of the smallest ratio and l that of the largest one. Note that q_s/p_s < 1 and that q_l/p_l > 1 because ∑q_i = ∑p_i = 1. Let D^−∞ and D^∞ be the values of D for α = −∞ and α = ∞, respectively. We find that D^−∞ = q_s/p_s and D^∞ = q_l/p_l. Finally, we have

Theorem 1. For any α-divergence and s and l such that q_s/p_s ≤ q_i/p_i ≤ q_l/p_l for all i,

Both bounds are in [0, 1]. The lower bound is larger than 0 unless q_s = 0, and the upper bound is less than 1 unless p_l = 0 (which is excluded by assumption). When q_s/p_s and q_l/p_l both go to 1, that is, when q_i goes to p_i for every i, the bounds become increasingly tight and, in the limit, become equal to Cr₀(¬A).

These bounds naturally extend to bounds on Cr(AB_i) = q_iCr(A). To simplify the notation, define r_i = q_i/p_i. We find that

In contrast with the bounds on Cr(¬A), which straddle the prior credence Cr₀(¬A), for some indices, the bounds on Cr(AB_i) exclude the corresponding prior credence. In particular, the lower bound exceeds Cr₀(AB_i) when r_i − r_l > Cr₀(A)(1 − r_l), and the upper bound is less than Cr₀(AB_i) when r_i − r_s < Cr₀(A)(1 − r_s). Since r_s < 1 < r_l, we find that Cr(AB_s) is less than Cr₀(AB_s) and Cr(AB_l) is larger than Cr₀(AB_l) for any α.

For the original Judy Benjamin problem, the range of possible credence values of Cr(B) is [1/3, 3/5], as was already found by Uffink (Reference Uffink1995). That for Cr(¬BH) is [3/10, 1/2], and that for Cr(¬B¬H) is [1/10, 1/6], neither one of which contains the prior value 1/4. In other words, even though Judy’s credence that she is in the Red Army’s area might increase or decrease when she gets the information from the duty officer, her credence that she is now in the Red Army’s headquarters region definitely increases, and her credence that she is now in the Red Army’s second company region definitely decreases.

3. Indeterminate Updating

Different members of the family of α-divergences may lead to different posteriors in the Judy Benjamin case, but the correct update rule for that problem should be found among the members of that family. It would clearly be desirable to find the assumed single correct posterior, but I argue in this section that such a result is not likely to emerge. Instead, the correct solution is (a subset of) the set of all posteriors produced by the family of rules. I now present and defend this new solution of the Judy Benjamin problem and address some possible objections.

3.3. Some Possible Objections

I next consider some possible objections to this account. First, it is unclear why rational ways of determining solutions to the Judy Benjamin equation should be restricted to minimizing α-divergences or, for that matter, minimizing functionals at all. Minimizing functionals is plausible because the new information comes in the form of constraints on the posterior credence function, and it is then reasonable to look for such functions that (a) meet the constraints and (b) are otherwise not too far away, in some sense, from the prior credence function. I defended the use of the family of α-divergences when I introduced them, but there might be other suitable families of update mechanisms. Uffink (Reference Uffink1995) has shown that only α-divergences will meet the Shore and Johnson axioms, but, of course, that argument is only as strong as the justification of those axioms. But even if another family would turn out to be more appropriate, it is not implausible that the same results would emerge. After all, the sole role of the members of the family is to determine Cr(¬A). Once that value has been determined, calculating the posterior credence functions just uses Probability Kinematics. The sole role of the family as a whole is then only to given bounds on the possible values of Cr(¬A).

That this consideration is not mere speculation is demonstrated by the family of f-divergences (Ali and Silvey Reference Ali and Silvey1966; Csiszár and Shields Reference Csiszár and Shields2004):

in which the kernel f is a nonnegative function on (0, ∞) with an everywhere-defined, continuous, and monotonically increasing derivative g and is such that f(1) = g(1) = 0. Minimizing members of this family is a plausible way of obtaining posterior credence functions because many f-divergences have been proposed and studied in the literature as measures of distance (Liese and Vajda Reference Liese and Vajda2006) between different probability functions. Some examples are the relative entropy and the inverse relative entropy used by Douven and Romeijn (Reference Douven and Romeijn2011) to effect Adams conditioning.Footnote ¹ Others are the Hellinger distance and the Kagan distance (f(x) = (x − 1)²/x). Furthermore, since g is monotonically increasing and equal to 0 when its argument is equal to 1, f has a minimum in (0, ∞) only when its argument is equal to 1. This minimum is set to 0 by requiring that f(1) = 0. Consequently, U_f is nonnegative and equal to 0 only when Cr(B_i) = Cr₀(B_i) for all i. Finally, U_f gets penalized, so to speak, increasingly severely for increasing deviations of Cr from Cr₀ because f is a convex function of x.

This family is different from that of the α-divergences. Nevertheless, the conclusions that can be drawn from both families are practically the same. If the new information is of the Jeffrey type, Probability Kinematics results, regardless of which f-divergence was used. The Judy Benjamin equation for the calculation of Cr(¬A) is replaced by a nonlinear equation whose solution is unique if it exists (app. A). Given Cr(¬A), the posterior credence function is provided by equation (11) for both α-divergences and f-divergences. The most important result, however, is that f-divergences, too, have nontrivial bounds on the possible values of Cr(¬A) and that those bounds are the same as those for α-divergences (app. B).

Second, it is disconcerting, to say the least, that precise information and a determinate prior can lead to an indeterminate posterior. I share this unease, but an indeterminate posterior does not imply that the agent is now licensed to hold her beliefs with whatever strengths she desires. Notice, for example, that the posterior is not indeterminate for all beliefs. If some belief is independent of the members of the Judy Benjamin partition, its posterior credence remains determinate and equal to the prior credence. Furthermore, even if the posterior credence becomes indeterminate, it may still contain nontrivial information. In the original Judy Benjamin case, for example, Judy’s credence that she is now in the headquarters area of the Red Army does increase in the sense that the lower bound on her posterior credence is larger than her prior credence that she is in that area.

The phenomenon of a determinate credence function becoming (partially) indeterminate is analogous to that of dilation (Seidenfeld and Wasserman Reference Seidenfeld and Wasserman1993). In the latter case, an already indeterminate prior becomes even more indeterminate, in a precisely defined manner, upon the acquisition of new information. In the coin toss experiment (Seidenfeld and Wasserman Reference Seidenfeld and Wasserman1993), for example, two coins are fair but the conjunction of both coins showing heads is indeterminate (between 0 and 0.5). Updating on Judy Benjamin type of information is more interesting because the prior credence is not indeterminate (or, at least, need not be), while the posterior may be, depending on the details of the background story. But just as classical dilation has a cause—indeterminate priors—the Judy Benjamin analogue to dilation has a cause as well: indeterminate updating. In that sense, the present phenomenon of Judy Benjamin type of information giving rise to an indeterminate posterior is simply another form of dilation.

4. Beyond Judy Benjamin

The standard credential update process is one in which the prior credence function is updated to a posterior credence function upon the acquisition of some new information. The various examples presented in this article make it clear that a single update mechanism cannot cover all subtleties and varieties contained in Judy Benjamin type of information. The only appropriate way then of dealing with that type of information is to consider all possible update mechanisms or, at least, all members of some large family of mechanisms that can plausibly be used to determine posterior credences. The interpretation defended in this article is to consider the indeterminateness fundamental: a given Judy Benjamin problem may have additional information that can be used to restrict the family of mechanisms, but, in general, it does not. On this view, the correct posterior in the presence of Judy Benjamin information is the set of posteriors obtained by employing (a subset of) all mechanisms in the family of plausible update mechanisms. The suggestion made in this article is that the family of update mechanisms is that of minimizing α-divergences, even though, as I mentioned in the preceding section, other families, such as that of f-divergences, might also be used.

We might ask why Judy Benjamin type of information is special as compared to Jeffrey type of information in that the former requires minimizing all divergences, leading to a set of posterior credence functions, while Probability Kinematics (or Bayesian Conditionalization for that matter) produces a single such function. But there really is nothing special. We could, and I propose we should, insist that modifying credences upon the delivery of new information is always done (at least, when the new information is a convex constraint on the posterior credence function) by minimizing all members of the family of divergences under the constraints appropriate to the new information—in other words, that updating always leads to a posterior set of credence functions. Minimizing all members of the family would constitute a standard credential update process if they all produced the same posterior credence function. It does so for Jeffrey information but not for Judy Benjamin information.

The Judy Benjamin problem as given by van Fraassen has served as the prime example of why minimizing the relative entropy cannot always be considered to be the correct way of determining a posterior credence function. That minimizing the relative entropy in the presence of Jeffrey information leads to Probability Kinematics is not a sufficient rational for continuing to use that functional exclusively, because any α-divergence (or f-divergence, for that matter) produces the same result. It is to be expected that minimizing the relative entropy for even more general types of information than the Judy Benjamin one may also produce implausible posterior credences in suitably chosen examples and that the general problem of constructing a posterior credence when new information of any type becomes available might better be done by constructing a set of posterior credence functions.

As a very general type of new information, we might, for example, consider posterior expectations of simple random variables, where “simple” means that the random variable has only a finite number of possible values. Jeffrey and Judy Benjamin types of information are a special case of that general form. The general solution proceeds by first determining Cr(.) for all members of the partition (as defined by the finite set of possible values of the random variable) by minimizing each member of the family of functionals and then using Probability Kinematics to obtain the general posterior credence set. As in the Judy Benjamin case, additional information contained in the accompanying story may reduce the set of functionals to be minimized, but, in general, the result will be a range of possible credence values for each member of the partition. Such a construction replaces updating by minimizing the relative entropy and does not suffer the limitations of the latter because it uses all possible members of the family and not just the single one whose choice has been motivated largely by past successes in restricted areas of inquiry.