Bayesian updating when what take yourself to learn might be false∗ Richard Pettigrew May 10, 2020 Abstract Michael Rescorla (2020) has recently pointed out that the standard arguments for Bayesian Conditionalization assume that whenever I take myself to learn something, it is true. Most people would reject this assumption. In response, Rescorla offers an improved Dutch Book argument for Bayesian Conditionalization that does not make this assumption. My purpose in this paper is two-fold. First, I want to illuminate Rescorla's new argument by giving a very general Dutch Book argument that applies to many cases of updating beyond those covered by Conditionalization, and then showing how Rescorla's version follows as a special case of that. Second, I want to show how to generalise Briggs and Pettigrew's Accuracy Dominance argument to avoid the assumption that Rescorla has identified (Briggs & Pettigrew, 2018). Careful formulations of the Bayesian norm of Conditionalization acknowledge that it governs how you should plan to update your credences, or how you should be disposed to update them. It does not govern how you should in fact update, or at least not directly. That is, Conditionalization does not govern the relationship between your prior credences and your posterior credences, but rather the relationship between your prior credences and your plans or dispositions for updating those priors. In particular, it governs those plans or dispositions you have for updating your credences in response to a certain sort of learning situation, namely, one in which you become certain of a proposition. An updating plan comprises two components: (i) a partition of the space of possibilities-if we think of the updating rule as describing a disposition, this partition specifies the possible stimuli; ∗Thank you to Michael Rescorla for extremely helpful comments on an earlier draft of this material. 1 (ii) a function that assigns to each element of that partition the posterior credence function that this plan endorses as a response to that element-if the updating rule describes a disposition, this partition matches each stimulus with its manifestation. When we consider Conditionalization, and we think of someone with a prior credence function c at time t, who is planning how to respond to evidence they receive between t and a later time t′ that comes in the form of a proposition they take themselves to have learned with certainty, the partition consists of the propositions: I take myself to learn E1 with certainty and nothing stronger by t′. I take myself to learn E2 with certainty and nothing stronger by t′. ... I take myself to learn En with certainty and nothing stronger by t′. We assume that the partition is finite. And the updating plan consists of conditional intentions or dispositions of the form: If I take myself to learn E1 with certainty and nothing stronger by t′, then I'll adopt credence function c1 at t′. If I take myself to learn E2 with certainty and nothing stronger by t′, then I'll adopt credence function c2 at t′. ... If I take myself to learn En with certainty and nothing stronger by t′, then I'll adopt credence function cn at t′. Throughout, I'll assume that E1, . . . , En are mutually exclusive.1 Conditionalization then says this: Conditionalization If (i) my credence function is c at t; (ii) c is defined on the algebra F ; (ii) my updating plan contains: If I take myself to learn Ei with certainty and nothing stronger by t′, then I'll adopt credence function ci at t′; and (iii) c(Ei) > 0; 1Rescorla (2020) assumes that they are exhaustive as well as exclusive, but it turns out that isn't necessary. 2 then rationality requires that, for all X in F , ci(X) = c(X|Ei) := c(XEi) c(Ei) In favour of this version of Conditionalization, there are at least four arguments: two pragmatic, two epistemic. The first pragmatic argument is David Lewis's diachronic Dutch Book argument (Lewis, 1999); the second is Peter M. Brown's expected utility argument (Brown, 1976); the first epistemic argument is Hilary Greaves and David Wallace's expected accuracy argument (Oddie, 1997; Greaves & Wallace, 2006); the second is R. A. Briggs and Richard Pettigrew's Accuracy Dominance argument (Briggs & Pettigrew, 2018). However, as Michael Rescorla (2020) has recently noted, each of these arguments relies on an assumption, namely, that the strongest proposition you take yourself to learn with certainty by t′ is true. That is, you are never wrong about what you take yourself to have learned-if I take myself to have learned that it is raining, then it is raining. As a result, if you think that this assumption doesn't always hold, then these arguments don't establish Conditionalization in all cases. They establish it only in those cases where the assumption holds. Rescorla offers an improved Dutch Book argument for Conditionalization that does not assume that, if you take yourself to learn Ei, then Ei must be true. My purpose in this paper is two-fold. First, I want to illuminate Rescorla's new argument by giving a very general Dutch Book argument that applies to many cases of updating beyond those covered by Conditionalization, and then showing how Rescorla's version follows as a special case of that. Second, I want to show how to generalise Briggs and Pettigrew's Accuracy Dominance argument to avoid the assumption that Rescorla has identified. 1 Two general norms connecting priors and possible posteriors Van Fraassen's original version of his Reflection Principle governs only your credence function at a particular time. It says that your credence in X at that time, conditional on your future credence in X being r, should be r (van Fraassen, 1984). Later, van Fraassen offered a different version- what Jonathan Weisberg calls General Credence Reflection-which says that your current credence in X should lie in the span of your possible future credences in X (van Fraassen, 1995, 1999; Weisberg, 2007). This governs the relationship between your current credences and your possible future credences. I want to consider a slightly stronger principle that governs that relationship. It says that your current credence function should 3 lie in the span of your possible future credence functions. I'll call it the General Reflection Principle. Van Fraassen's original Reflection Principle follows from the General Reflection Principle if each future credence function is certain that it is the future credence function (Weisberg, 2007). What are your possible future credence functions? There are at least two ways to spell out the modality here: one epistemic, the other ontic. Van Fraassen and Weisberg give epistemic readings: for them, a credence function is possible if it is foreseen or foreseeable. Thus, you might say that, at time t, c′ is a possible credence function at t′ if it is epistemically possible that you'll have a particular learning experience between t and t′ and you plan to respond to that experience by adopting c′. On the other hand, you might take an ontic approach. Thus, you might say that, at time t, c′ is a possible credence function at t′ if it is metaphysically or physically or nomologically possible that you'll have a particular learning experience between t and t′ and you are disposed to respond to that by adopting c′. The arguments I present here are agnostic between these two readings. They will work whichever interests you. The General Reflection Principle is a more general norm than Conditionalization. Conditionalization covers cases in which you arrive at your future credence function in a particular way. It applies when you take yourself to learn something with certainty and set your future credence function by using your updating rule to respond to that learning experience. The principle that interests me covers those cases and other cases besides: cases when you take your evidence to shift your opinion without teaching you anything with certainty, such as in Richard Jeffrey's case of viewing cloth by candlelight or van Fraassen's Judy Benjamin case (Jeffrey, 1965; van Fraassen, 1981). There is a weaker and a stronger way to make the General Reflection Principle. First, the weak one. Suppose c is your credence function at t, and R is the set of credence functions you might have at time t′. That is, c is your prior and R contains all and only your possible posteriors. We'll assume throughout that R is finite. Then the Weak General Reflection Principle says that c should be in the convex hull of R.2 That is, c should be a convex combination or weighted sum or weighted average of the credence functions in R. That is: Weak General Reflection Principle Suppose c is your credence function at t, and R = {c1, . . . , cn} is the set of credence functions you might have at t′. Then rationality requires that there is, for each ci in R, a non-negative weighting 0 ≤ λi ≤ 1 such 2The convex hull of a set is the smallest convex set that contains it. A set is convex when it contains any mixture of any two elements it contains. Thus, the interior of a circle is convex, but its circumference is not. 4 that ∑ni=1 λi = 1 and c(−) = n ∑ i=1 λici(−) And the Strong General Reflection Principle says that your prior should be in the interior of the convex hull of your possible posteriors. That is, Strong General Reflection Principle Suppose c is your credence function at t, and R = {c1, . . . , cn} is the set of credence functions you might have at t′. Then rationality requires that there is, for each ci in R, a positive weighting 0 < λi < 1 such that ∑ni=1 λi = 1 and c(−) = n ∑ i=1 λici(−) 2 From General Reflection to Conditionalization As we will see, we can establish Weak GRP via Dutch Book or Accuracy Dominance arguments; and we can establish Strong GRP via a Dutch Book argument, but not via an Accuracy Dominance argument. In this section, I show that, from either of these we can obtain Conditionalization. Throughout, we will assume that all credence functions are defined on the same finite algebra of propositions, F . Now, suppose first that each of your possible future credence functions are the result of following your updating rule. The partition on which your updating rule is defined consists of the following propositions: I take myself to learn Ei with certainty and nothing stronger by t′, for i = 1, . . . , n, where each Ei is in F . And the rule itself consists of the following conditionals: If I take myself to learn Ei with certainty and nothing stronger by t′, then I'll adopt credence function ci at t′, for i = 1, . . . , n. So R = {c1, . . . , cn}. Now, suppose that (i) E1, . . . , En are mutually exclusive propositions, and (ii) ci(Ei) = 1, for i = 1, . . . , n. 5 Then, for j = 1, . . . , n, c(Ej) = ∑ni=1 λici(Ej) by Weak GRP = λjcj(Ej) since E1, . . . , En are exclusive, so ci(Ej) = 0 if i 6= j = λj since cj(Ej) = 1 Thus, for all X in F and Ej c(XEj) = ∑ni=1 λici(XEj) by Weak GRP = ∑ni=1 c(Ej)ci(XEj) by above = c(Ej)cj(XEj) since ci(XEj) = 0 if i 6= j = c(Ej)cj(X) since cj(Ej) = 1 Thus, if c(Ej) > 0, then cj(X) = c(X|Ej), as required. 2 Note that, at no point did we assume that you take yourself to learn Ej only if Ej is true. 3 The Dutch Book argument for GRP In the Dutch Book argument for Probabilism, we show that, if your credence function does not obey the axioms of the probability calculus, there is a series of bets, each of which your credences require you to take, but which, when added together, lose you money for sure. In the traditional Dutch Book argument for Conditionalization, we show that, if you plan to update your prior in some way other than by Conditionalization, then there is a series of bets, each of which your priors require you to take, and, for each of your possible posteriors after applying the updating rule, there is a series of bets, each of which those posteriors require you to take, but which, when added together, lose you money for sure. Rescorla's improved Dutch Book argument for Conditionalization does not require a different series of bets for each possible posterior; rather, there is a series of bets, each of which your priors require you to take, and there is a series of bets, each of which all your possible posteriors require you to take, but which, when added together, lose you money for sure. The same is true of the Dutch Book arguments for Weak and Strong GRP. First, let's specify two different ways in which a prior credence function and a set of possible posterior credence functions, taken together, might be flawed; the Dutch Book arguments will take some subset of these flaws to 6 indicate irrationality. Instead of talking about bets specifically, we'll talk generally about any sort of decision-theoretic act. Some terminology: • A possible world is a classically consistent assignment of truth values to the propositions on which the individual's credence function is defined-that is, the propositions in F . We denote the set of possible worlds W . Since F is a finite algebra, for each possible world w inW , there is a unique proposition in F that is true at that world and no other. We abuse notation and write w also for the proposition. • An act is a function that assigns to each possible world the amount of utility you receive in that world (where all utility in what follows is measured on the same scale). In the language of probability theory, an act is a random variable. • If A is an act and w is a possible world, A(w) is the amount of utility you receive if you choose A at w. • If m is a utility level, then m is the constant act that takes that utility at every possible world; that is, m(w) = m, for all worlds w. • A decision problem is a set of available acts. • If A belongs to one decision problem and A′ belongs to another, then A + A′ is the act of choosing A from the first and A′ from the second and its utility at w is (A + A′)(w). • If c is a probabilistic credence function c and A and B are acts, c prefers A to B if the expected value of A relative to c is strictly greater than the expected value of B relative to c. That is, ∑ w c(w)A(w) > ∑ w c(w)B(w) • If A and B are acts, – A strongly dominates B if A(w) > B(w) for all possible worlds w; – A weakly dominates B if (i) A(w) ≥ B(w) for all possible worlds w, and (ii) A(w) > B(w) for some possible worlds w. Suppose c is your credence function at t and R is the set of credence functions you might have at t′. Weak Dutch Strategy (c, R) is vulnerable to a weak Dutch strategy if there are acts A, B, A′, and B′ such that (i) c prefers A to B; (ii) each ci in R prefers A′ to B′ 7 (iii) B + B′ weakly dominates A + A′. Strong Dutch Strategy (c, R) is vulnerable to a strong Dutch strategy if there are acts A, B, A′, and B′ such that (i) c prefers A to B; (ii) each ci in R prefers A′ to B′ (iii) B + B′ strongly dominates A + A′. We can now state our Dutch Book theorem for Weak and Strong GRP: Theorem 1 (I) (c, R) violates Weak GRP iff it is vulnerable to a strong Dutch strategy. (II) (c, R) violates Strong GRP iff it is vulnerable to a weak Dutch strategy. I will now set out of the proof of Theorem 1(I). As so often in Dutch book arguments, it relies on the Separating Hyperplane Theorem. The proof of Theorem 1(II) is very similar, but appeals to the Supporting Hyperplane Theorem instead. In this context, the Separating Hyperplane Theorem says the following: Suppose c is not in the convex hull of R, as is the case when (c, R) violates Weak GRP. Then there is an act A such that, for any ci in R, ∑ w∈W ci(w)A(w) < ∑ w∈W c(w)A(w) Therefore, there are two numbers, m and n, such that, for any ci in R, ∑ w∈W ci(w)A(w) < m < n < ∑ w∈W c(w)A(w) So • n < ∑w∈W c(w)A(w); and • −m < ∑w∈W ci(w)(−A(w)), for i = 1, . . . , n. Then c prefers A to the constant act n, while each ci prefers −A to the constant act −m. But (A + (−A))(w) = 0 < n−m = (n +−m)(w) That is, n +−m strongly dominates A + (−A). Thus, (c, R) is vulnerable to a strong Dutch strategy, as required. Next, suppose (c, R) does satisfy Weak GRP. Then c = ∑i λici. Now, since c prefers A to B and ci prefers A′ to B′, • ∑w c(w)B(w) < ∑w c(w)A(w) 8 • ∑w ci(w)B′(w) < ∑w ci(w)A′(w), for i = 1, . . . , n. Then ∑ w,i λici(w)(B + B′)(w) = ∑ w,i λici(w)B(w) + ∑ w,i λici(w)B′(w) = ∑ w c(w)B(w) + ∑ w,i λici(w)B′(w) since c = ∑i λici = ∑ w c(w)B(w) + ∑ i λi ∑ w ci(w)B′(w) < ∑ w c(w)A(w) + ∑ i λi ∑ w ci(w)A′(w) = ∑ w c(w)A(w) + ∑ w,i λici(w)A′(w) since c = ∑i λici = ∑ w,i λici(w)A(w) + ∑ w,i λici(w)A′(w) = ∑ w,i λici(w)(A + A′)(w) So it cannot be that (A + A′)(w) < (B + B′)(w) for all w. 2 4 The Accuracy Dominance argument for GRP In the Accuracy Dominance argument for Probabilism, we show that, if your credence function does not obey the axioms of the probability calculus, there is an alternative credence function, defined on exactly the same propositions, that has greater accuracy than yours for sure. In Briggs and Pettigrew's Accuracy Dominance argument for Conditionalization, we show that, if you plan to update your prior in some way other than by Conditionalization, then there is an alternative prior and an alternative updating rule that have greater total accuracy than your prior and updating rule for sure. In the Accuracy Dominance argument for Weak GRP, we show that, if your prior is not a convex combination of your possible posteriors, there is an alternative prior and alternative posteriors, each paired with one of your actual posteriors, such that if you were to replace your prior with the alternative prior and each of the posteriors with its paired alternative, you'd have greater total accuracy for sure. First, a couple of quick words about measuring accuracy. Briggs and Pettigrew's argument, like the increasingly standard Accuracy Dominance argument for Probabilism, assumes that our measures of inaccuracy have three properties: 9 Additivity The inaccuracy of a whole credence function is the sum of the inaccuracy of the credences it assigns. More precisely: If I is a legitimate measure of the inaccuracy of a credence function at a world, then there is, for each X in F , a scoring rule sX : {0, 1} × [0, 1] → [0, ∞] such that, for any credence function c defined on F and any world w, I(c, w) = ∑ X∈F sX(vw(X), c(X)) where vw(X) = 0 if X is false at w and vw(X) = 1 if X is true at w. In this case, we say that s generates I. Continuity The inaccuracy of a credence is a continuous function of that credence. More precisely: If I is a legitimate measure of inaccuracy and I is generated by s, then, for all X in F , sX(1, x) and sX(0, x) are continuous functions of x. Strict Propriety Each credence expects itself to be most accurate. More precisely: If I is a legitimate measure of inaccuracy and I is generated by s, then, for all X in F and 0 ≤ p ≤ 1, psX(1, x) + (1− p)sX(0, x) is uniquely minimized, as a function of x, at x = p. When I satisfies these three properties, we say that it is an additive and continuous strictly proper inaccuracy measure. Next, let's specify a flaw that a prior and set of possible posteriors might jointly have. Accuracy Domination (c, R) is accuracy dominated iff, for all legitimate inaccuracy measures I, there is (c?, R?) such that, for all possible worlds w and all i = 1, . . . , n, I(c?, w) + I(c?i , w) < I(c, w) + I(ci, w) And now our theorem: Theorem 2 (c, R) violates Weak GRP iff it is accuracy dominated. I will now set out the proof of Theorem 2. Given a set of possible posteriors, R, define the following set of (n + 1)-dimensional vectors of credence functions: R = {(vw, c1, . . . , ci−1, vw, ci+1, . . . , cn) : w ∈ W , i = 1, . . . , n} 10 where, again, vw(X) = 0 if X is false at w and vw(X) = 1 if X is true at w. Then we show that, if c violates Weak GRP, then (c, c1, . . . , cn) 6∈ R + To prove this, we prove the contrapositive. Suppose (c, c1, . . . , cn) ∈ R +. Then there are 0 ≤ λw,i ≤ 1 such that ∑w ∑i λw,i = 1 and (c, c1, . . . , cn) = ∑ w ∑ i λw,i(vw, c1, . . . , ci−1, vw, ci+1, . . . , cn) Thus, c = ∑ w ∑ i λw,ivw and cj = ∑ w λw,jvw + ∑ w ∑ i 6=j λw,icj So (∑ w λw,j)cj = ∑ w λw,jvw So let λj = ∑w λw,j. Then, for j = 1, . . . , n, λjcj = ∑ w λw,jvw And thus ∑ i λici = ∑ i ∑ w λw,ivw = c Thus, c is in R+, and c satisfies Weak GRP. Now, we appeal to two central facts about additive and continuous strictly proper inaccuracy measures: Lemma 3 (Proposition 2, (Predd et al., 2009)) Suppose I is an additive and continuous strictly proper inaccuracy measure. Then there is a Bregman divergence D such that, for any credence function and any world,3 I(c, w) = D(vw, c) Lemma 4 (Proposition 3, (Predd et al., 2009)) Suppose D is a Bregman divergence and P is a set of credence functions. Then, if c is not in P+, then there is c? in P+ such that, for all p in P , D(p, c?) < D(p, c) 3A Bregman divergence is a certain sort of function that takes pairs of credence functions defined on F and returns a non-negative real number or infinity. It might be thought of as measuring the distance from one credence function to the other, but it lacks certain basic properties of a measure of distance. 11 Now, suppose I is an additive and continuous strictly proper inaccuracy measure and DI is its accompanying Bregman divergence. Then (c, c1, . . . , cn) 6∈ R + Then there is (c?, c?1 , . . . , c ? n) ∈ R + such that, for all w and i, D(vw, c?) +D(c1, c?1) + . . . +D(ci−1, c ? i−1)+ D(vw, c?i ) +D(ci+1, c ? i+1) + . . . +D(cn, c ? n) < D(vw, c) +D(c1, c1) + . . . +D(ci−1, ci−1)+ D(vw, ci) +D(ci+1, ci+1) + . . . +D(cn, cn) Now, D(ci, c?i ) ≥ 0 and D(ci, ci) = 0. So we can infer: D(vw, c?) +D(vw, c?i ) < D(vw, c) +D(vw, ci) And thus I(c?, w) + I(c?i , w) < I(c, w) + I(ci, w) as required. Next, suppose (c, R) does satisfy Weak GRP. Then c = ∑ i λici Now consider c?, c?1 , . . . , c ? n. Then, since I is strictly proper: • ∑w c(w)I(c, w) ≤ ∑w c(w)I(c?, w) • ∑w ci(w)I(ci, w) ≤ ∑w c(w)I(c?i , w), for i = 1, . . . , n. Then ∑ w,i λici(w)(I(c, w) + I(ci, w)) = ∑ w,i λici(w)I(c, w) + ∑ w,i λici(w)I(ci, w) = ∑ w c(w)I(c, w) + ∑ w,i λici(w)I(ci, w) since c = ∑i λici = ∑ w c(w)I(c, w) + ∑ i λi ∑ w ci(w)I(ci, w) ≤ ∑ w c(w)I(c?, w) + ∑ i λi ∑ w ci(w)I(c?i , w) = ∑ w c(w)I(c?, w) + ∑ w,i λici(w)I(c?i , w) = ∑ w,i λici(w)I(c?, w) + ∑ w,i λici(w)I(c?i , w) since c = ∑i λici = ∑ w,i λici(w)(I(c?, w) + I(c?i , w)) 12 So it cannot be that I(c?, w) + I(c?i , w) < I(c, w) + I(ci, w) for all w inW . 2 5 Problems with Reflection We are left, then, with two arguments for Weak GRP (and one for Strong GRP). If you violate Weak GRP, your prior and possible posteriors are jointly flawed in two ways. First, they require you to choose a dominated pair of acts in certain situations. Second, there are alternatives that accuracy dominate you. Provided you agree that those flaws render your priors and possible posteriors jointly irrational, we have two arguments for Weak GRP, one epistemic, one pragmatic. And, as we noted above, with Weak GRP in place, we can derive Conditionalization without assuming that anything we take ourselves to learn is true. However, a natural worry arises. There are countless putative counterexamples to van Fraassen's original formulation of the Reflection Principle (Talbott, 1991; Christensen, 1991; Arntzenius, 2003; Briggs, 2009). Do they not tell equally against the formulations we've given here? I think not, though as I'll explain I'm less than fully certain. On Van Fraassen's original formulation, the principle governs only your prior credence function. It imposes no constraints on your future credences, but only on the relationship between your prior opinions about your future credences in certain matters and your prior credences in those matters. It says that your current credence in X, conditional on your future credence in X being r, should be r. Let's consider one of the counterexamples due to David Christensen (1991). I know that, between t and t′, I will learn no new evidence, but I will begin to feel the effects of a hallucinogenic drug that I've just taken. I also know that one of the effects is that it will make me very confident that I can fly. The Reflection Principle deems irrational a credence function that is simultaneously certain that my future credence that I can fly will be high, and very confident that I cannot fly. But that seems a perfectly rational response to the evidence I currently have. What does the Weak GRP say about this case? It says that, if my only possible future credence function is very confident that I can fly, while my current credence function is very confident I cannot, then they are jointly irrational. But that isn't a problem. After all, I'd also agree that they are jointly irrational, because I judge my future credences to be irrational- they're the result of taking a hallucinogen, and my aerial competence is not one of the truths typically revealed by such drugs. The upshot: this counterexample doesn't tell equally against Weak GRP. The reason is that the original Reflection Principle is a purely synchronic norm, while Weak GRP is genuinely diachronic. 13 This might lead you to wonder how Weak GRP can help to establish Conditionalization, which I noted at the beginning is a synchronic norm governing the relationship between your credence function at t and your updating plan at t, not between your credence function at t and your possible credence functions at t′. The reason is that an updating rule is a commitment to having certain credence functions at t′ given certain learning experiences prior to that. So it is reasonable to judge it by looking at what would happen were you to follow that plan and update accordingly. If you were to do that, then you'd have certain possible posteriors. And if your prior together with those possible posteriors are irrational because they violate Weak GRP, then your prior together with the updating rule that demands those posteriors is similarly irrational. However, a different sort of counterexample does concern me. Suppose you are a permissivist about rationality. You have a certain body of evidence that permits two different credence functions c and c′. In the evening, at time t, your credence function is c-perhaps in the evening, after the stress of a long day, you're more inclined to the more extreme of the two permissible responses. But, from past experience, you know that, by the morning, after a long sleep during which you will gain no new evidence, your credence function will be c′-perhaps you know that, after some rest, the stress will dissipate somewhat and you'll instead favour the less extreme permissible response. So c′ is your only possible future credence function. But it is not c. So, together, they violate Weak GRP. Unlike the case of the hallucinogenic drug above, in this case both credence functions are rationally permissible. So, if they are jointly irrational, their individual irrationality cannot be the reason. Instead, it must lie in the move from one to the other. I find myself without a clear sense of whether Weak GRP gives the correct verdict in this case. In the drug case, I had a strong conviction that the prior credence function that violates Reflection is perfectly rational. So in that case Reflection was in direct conflict with my judgments. 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