anal_641-655.fm Analysis 67.1, January 2007, pp. 50–59. © Terry Horgan Blackwell Publishing Ltd.Oxford, UK and Malden, USAANALAnalysis0003-26382007 Blackwell Publishing Ltd.January 20076715059ArticlesTerry Horgan SYNCHRONIC BAYESIAN UPDATING Synchronic Bayesian updating and the generalized Sleeping Beauty problem Terry Horgan The two principal answers defended in the literature on the Sleeping Beauty problem (Elga 2000) are 1/2 and 1/3. Roger White (2006) poses a generalized version of the problem. ‘When the main arguments for the answer 1/3 are extended to the generalized case,’ he maintains, ‘they have an unacceptable consequence, whereas extending the halfer’s reasoning turns out rather nicely’ (114). Here I will argue that although my favoured reasoning for the 1/3 answer (Horgan 2004, in press) does have the consequence in question, this consequence is quite acceptable given my treatment of the original problem. Any apparent unacceptability the consequence initially might seem to possess, I claim, results from a failure to appreciate the nature of newly available infor- mation in the Sleeping Beauty problem – in both the original version and White’s generalized version. Once this (easily overlooked) new information and its evidential relevance are taken into account – as is done in my treatment of the original problem – the allegedly unaccept- able consequence, in the case of the generalized problem, can be seen to be unobjectionable. Some thirders (e.g. Elga) disagree with me about the presence of new information in the Sleeping Beauty problem; they deny that the correct answer in the original problem turns on the acquisition of new informa- tion. I will argue that White’s generalized Sleeping Beauty problem poses a genuine difficulty for those thirders, even though my own version of thirdism handles the problem straightforwardly. 1. The original Sleeping Beauty problem, White’s generalized version, and White’s challenge The original Sleeping Beauty problem goes as follows. On Sunday Sleeping Beauty learns that she will be put into dreamless sleep for the next two days in a sleep laboratory. She will be awakened briefly on Monday by the experimenters, and then returned to dreamless sleep. If a fair coin that is to be tossed prior to Monday evening lands Heads, then she will sleep through until Wednesday and will awaken by herself. If the coin comes up Tails, then on Monday evening her memory of the Monday awakening will be erased, and she will be briefly awakened again on Tuesday morning by the experimenters. When she is awakened on Monday, with no memory synchronic bayesian updating 51 of a prior awakening, what probability should she assign to the proposi- tion that the coin lands Heads? (One version has the coin toss occurring before the Monday awakening; another has it occurring afterwards. Most who have discussed the problem, including both White and myself, main- tain that it doesn’t matter.) White poses the following generalized version of the problem, involving a random waking device with an adjustable randomizer: A random waking device has an adjustable chance c ∈ (0,1] of wak- ing Sleeping Beauty when activated on an occasion. In those circum- stances in the original story where Beauty was awakened, we now suppose only that this waking device is activated. When c = 1, we have the original Sleeping Beauty problem. But if c < 1, the case is significantly different. For in this case Beauty cannot be sure in advance that she will be awakened at all during the experiment. When she does wake up she clearly gains some relevant information. (116) White’s challenge to thirders has two parts. First, he claims that their arguments for the 1/3 answer in the original problem also apply directly to the generalized version, regardless of the setting of the parameter c. (He defends this claim in the body of his paper vis-à-vis Elga’s argument and an argument by Dorr (2002) and Arntzenius (2003), and in note 2 vis-à-vis my argument.) Second, he reasons as follows about the Dorr- Arntzenius arguments (and about mine, as note 2 makes explicit): So according to the Elga and Arntzenius-Dorr [and Horgan] argu- ments, then, the introduction of the variable c has no effect on the answer to the problem. But this, I submit, cannot be right. As we have noted, if c < 1 then when Beauty wakes up she clearly does gain some information, namely W: Beauty is awake at least once during the experiment. And this is clearly relevant to whether H: The coin landed heads. For the likelihood of W is greater given ∼H than given H. Any answer must take into account the impact of this information on Beauty’s credence. For the difference between the likelihoods P_(W|H) and P_(W|∼H) increases as c decreases (where P_ is Beauty’s rational credence function prior to waking). The degree to which Beauty has a better chance of being awakened given two opportunities rather than one depends on how small c is. So whatever else we might say about Beauty’s rational credence in H when she wakes up, it should vary to some degree with the value of c. This is the result that the 52 terry horgan thirder, insofar as he follows the Elga and Arntzenius-Dorr [or Hor- gan] arguments, cannot accommodate. (117–18, my italicization of the penultimate sentence) I will call the inference embodied in the italicized sentence the H-variabil- ity inference. 2. Synchronic Bayesian updating and the original Sleeping Beauty problem Let me focus first on the original Sleeping Beauty problem, before address- ing the generalized version and White’s challenge. We thirders agree with one another that in the original problem, for Beauty the epistemic prob- ability of H changes from 1/2 on Sunday to 1/3 when she is awakened on Monday. But we disagree about why this is so. Elga maintains that the epistemic probability changes even though Beauty gains no new relevant information upon being awakened. I maintain, on the contrary, that she does gain new relevant information – and that the change in epistemic probability of H results from what I will here call synchronic Bayesian updating on this information.1 Let me briefly rehearse the reasoning I recommend. When Beauty is awakened on Monday, she thereby acquires a piece of information she did not possess on Sunday – information she expresses with the indexical statement I am awakened today by the experimenters. This counts as relevant new information because one possibility about today that is consistent with her Sunday information – but not with her current total information – is H and today is Tuesday. (Were this how things actually are today, then of course she would currently be in dreamless asleep, and would not be pondering the problem. Nonetheless, it is a possibility about today that is consistent with what she knew on Sunday.) She considers the following four possibilities (where H and T are HEADS and TAILS respectively): TodayH,Mon: H and today is Monday. TodayH,Tues: H and today is Tuesday. TodayT,Mon: T and today is Monday. TodayT,Tues: T and today is Tuesday. 1 Weintraub (2004) explicitly agrees with me that Beauty acquires new relevant information upon being awakened by the experimenters, and about the nature of this information. Although Dorr (2002) and Arntzenius (2003) are not explicit on the matter, I believe that my own approach captures the underlying spirit of both of theirs. Bostrum (in press) claims, as I do, that Beauty acquires new information and that it is indexical; but his approach is otherwise very different. synchronic bayesian updating 53 She first assigns preliminary probabilities to these four possibilities – probabilities they possess relative to a certain proper subset of her total current information. The proper subset in question excludes her informa- tion that she has been awakened today by the experimenters.2 (But it does not exclude her information that today is either Monday or Tuesday.) These preliminary probabilities, expressed by ‘P_’, are: P_(TodayH,Mon) = 1/4 P_(TodayH,Tues) = 1/4 P_(TodayT,Mon) = 1/4 P_(TodayT,Tues) = 1/4 She also assigns the following preliminary conditional probabilities to these four possibilities, relative to the same proper subset of her total current information (and letting ‘AToday’ symbolize ‘I am awakened today by the experimenters’): P_(TodayH,Mon|AToday) = 1/3 P_(TodayH,Tues|AToday) = 0 P_(TodayT,Mon|AToday) = 1/3 P_(TodayT,Tues|AToday) = 1/3 She now does Bayesian updating by means of these preliminary condi- tional probabilities, thereby taking into account her additional current information that AToday. The resulting updated probabilities are: P(TodayH,Mon) = 1/3 P(TodayH,Tues) = 0 P(TodayT,Mon) = 1/3 P(TodayT,Tues) = 1/3 And so, The Bayesian updating she employs is synchronic, because the four pos- sibilities to which she assigns preliminary probability all have arisen post- awakening – by virtue of the fact that upon being awakened, she no longer knows which day it is (although she does know that today is either Monday or Tuesday). But it is Bayesian updating nonetheless, even though it does not conform to the familiar special case in which the preliminary 2 It also excludes her information that she is awake right now, since if right now were Tuesday and the coin had landed heads, then she would now be in dreamless sleep. P H P Today H,Mon( ) = ( ) = 1 3 P T P Today P TodayT,Mon T,Tues( ) = ( ) + ( )[ ] = 2 3 54 terry horgan probabilities are temporally prior probabilities.3 Accordingly, I will call my position Bayesan thirdism.4 3. Synchronic Bayesian updating and the generalized Sleeping Beauty problem Let me now explain how my recommended form of reasoning extends to White’s generalization of the Sleeping Beauty problem. With the variable parameter c in the picture, Beauty’s preliminary probabilities are still the same as before: Her preliminary probabilities for the pertinent conjunctions involving ‘AToday’, again bracketing the information about having been awakened today, are these: P_(TodayH,Mon & AToday) = 1/4c P_(TodayH,Tues & AToday) = 0 P_(TodayT,Mon & AToday) = 1/4c P_(TodayT,Tues & AToday) = 1/4c Thus, concerning the possibility that she is awakened today by the experi- menters, her preliminary probability is this: So, plugging into the definition of conditional probability, her preliminary conditional probabilities (with the awakening-information still bracketed) are: 3 It bears emphasis that, even in instances of the familiar special case, one’s Bayesian updating still employs preliminary probabilities that accrue to various possibilities relative to a proper subset of one’s total current information. In the special case, this proper subset happens to coincide with a body of information that was previ- ously one’s total pertinent information. 4 This label is intended to highlight the role of Bayesian updating in my treatment of the problem. Needless to say, one can advocate Bayesian thirdism without embracing the package of subjectivist views about the nature of epistemic probability, and about constraints on rational belief-formation, commonly called ‘Bayesianism’. I myself am no fan of Bayesianism, although I am a Bayesian thirder. P_ Today P Today P Today P Today H,Mon H,Tues T,Mon T,Tues ( ) = ( ) = ( ) = ( ) = _ _ _ 1 4 P A P Today A P Today A P Today A Today H,Mon Today T,Mon Today T,Tues Today _ _ & _ & _ & ( ) = ( ) + ( ) + ( ) = 3 4 c P Today A P Today A P AH,Mon Today H,Mon Today Today_ _ & _( ) = ( ) ( ) = ( ) ( ) =1 4 3 4 1 3c c synchronic bayesian updating 55 She now does Bayesian updating by means of these preliminary condi- tional probabilities, thereby taking into account her additional current information that AToday. The resulting updated probabilities, just as in the original problem, are: P(TodayH,Mon) = 1/3 P(TodayH,Tues) = 0 P(TodayT,Mon) = 1/3 P(TodayT,Tues) = 1/3 So again, just as in the original problem, Thus the original reasoning, employing synchronic Bayesian updating, extends straightforwardly to the generalized version of the problem. Beauty’s newly acquired information that she has been awakened today by the experimenters has the same effect in the general case as in the special case where c = 1: this information changes the epistemic probabil- ity of H, which was 1/2 on Sunday, to 1/3. 4. Bayesian thirdism and the H-variability inference: diagnosing the fallacy Against the background of the preceding discussion, let us now consider White’s H-variability inference. The possibility he labels W can be usefully reformulated as follows, without prejudice to his argument (and in a way that conforms with the first-person indexical language employed in AToday): A1: I am awakened by the experimenters at least once during Monday and Tuesday. Thus reformulated, the premiss of the inference is this: PREMISS: If c < 1 then P_(A1|∼H) > P_(A1|H); furthermore, ‘the degree to which Beauty has a better chance of being P Today A P Today A P AH,Tues Today H,Tues Today Today_ _ & _( ) = ( ) ( ) = ( ) =0 3 4 0c P Today A P Today A P AT,Mon Today T,Mon Today Today_ _ & _( ) = ( ) ( ) = ( ) ( ) =1 4 3 4 1 3c c P Today A P Today A P A c c T,Tues Today T,Tues Today Today_ _ & _( ) = ( ) ( ) = ( ) ( ) =1 4 3 4 1 3 P H P Today H,Mon( ) = ( ) = 1 3 P T P Today P TodayT,Mon T,Tues( ) = ( ) + ( )[ ] = 2 3 56 terry horgan awakened given two opportunities rather than one depends on how small c is’ (118). On my account, this premiss is certainly true. For, letting ‘AMon’ and ‘ATues’ respectively symbolize ‘I am awakened by the experimenters on Monday’ and ‘I am awakened by the experimenters on Tuesday’, Thus, when c = 1, while, as c approaches 0, the ratio of the quantity (2c − c 2) to the quantity c grows increasingly greater. White reasons as follows, on the basis of PREMISS: ‘So, whatever else we might say about Beauty’s rational credence in H when she wakes up, it should vary to some degree with the value of c’ (118). (This is what I above labelled the H-variability inference.) But, on my account, this infer- ence is mistaken. For, as argued already, the right way to assign epistemic probabilities to H and T is by means of synchronic Bayesian updating (using the information that AToday) on the following preliminary condi- tional probabilities: And the result of such updating is the same, regardless of the value of the parameter c: P A H P A A H P A H P A H P A & A H Mon Tues Mon Tues Mon Tues _ _ _ _ 1 0 0 ( ) = ( )( ) = ( ) + ( ) - ( )( ) = + - = v c c P A H P A T P A A T P A T P A T P A & A T c c Mon Tues Mon Tues Mon Tues _ ~ _ _ _ 1 1 22 ( ) = ( ) = ( )( ) = ( ) + ( ) - ( )( ) = + - ¥( ) = - v c c c c P A H_ 1 1( ) = P A ~H ,_ 1 2 1 1( ) = -( ) = P Today A P Today A P Today A H,Mon Today T,Mon Today T,Tues Today _ _ _( ) = ( ) = ( ) = 1 3 P Today AH,Tues Today_( ) = 0 P H P Today H,Mon( ) = ( ) = 1 3 P T P Today P TodayT,Mon T,Tues( ) = ( ) + ( )[ ] = 2 3 synchronic bayesian updating 57 What makes the H-invariability inference seem initially plausible, even though it is actually fallacious? Well, suppose that one believes – as many who have written about the Sleeping Beauty problem do believe – that, in the original version of the problem, Beauty obtains no new relevant information upon being awakened on Monday by the experimenters. Then one will also be strongly inclined to believe, concerning the gen- eralized Sleeping Beauty problem, that the strongest new relevant infor- mation Beauty receives on Monday is the information A1. And if one believes that, then White’s H-variability inference becomes extremely plausible. After all, White is right to claim that ‘the degree to which Beauty has a better chance of being awakened given two opportunities rather than one depends on how small c is’ (118). So, if indeed the strongest new relevant information Beauty receives is A1, then it becomes very hard to see how the information that A1 could fail to have a differentially stronger effect on the epistemic probability of H, depending on how small c is. Moreover, again on the assumption that A1 is the strongest new relevant information Beauty receives upon being awakened, the natural-looking way to take account of this information is to do diachronic Bayesian updating, by means of the preliminary probability P_(H|A1). (The updat- ing is diachronic because A1 is a state of affairs expressible without any temporal-indexical term like ‘today’, and thus P_(H|A1) has the same numerical value on Sunday that it has when Beauty is awakened on Monday.) By Bayes’s theorem, as White shows, As c → 1, P_(H|A1) → 1/2 As c → 0, P_(H|A1) → 1/3 Thus, diachronic Bayesian updating via P_(H|A1) gives these results: As c → 1, P(H) → 1/2 (the halfer’s answer to the original problem) As c → 0, P(H) → 1/3 (the thirder’s answer to the original problem) So White’s H-variability inference, and also his use of diachronic Bayesian updating via the preliminary conditional probability P_(H|A1), are both very plausible and natural, given the assumption that the strongest newly acquired relevant information, in the generalized Sleeping Beauty Prob- lem, is A1. 5 But on my account this assumption is false – as is the assumption’s consequence that in the original problem no new relevant information is 5 The presence of this assumption in White’s reasoning surfaces explicitly in the following passage: ‘Halfers are suspicious of any shift in credence that is not in response to new relevant information. So in the generalized case they insist that Beauty should simply update her credence in the standard way by conditionalizing on her strongest new information, namely W’ (118, my italics – where W = ‘Beauty is awake at least once during the experiment’). 58 terry horgan acquired. Upon being awakened by the experimenters, Beauty actually acquires not merely the information that she is awakened at least once by the experimenters, but also the stronger (more specific, essentially index- ical) information that she is awakened today by the experimenters. (The latter piece of information entails the former under the conditions of the problem, but not conversely.) Thus, the fallacy in the H-variability infer- ence, and in diachronic Bayesian updating via P_(H|A1), is that they ignore relevant new information. The correctly updated epistemic probability for H is obtained by conditionalizing not on the partial new relevant information A1, but rather on the strongest new relevant information – viz., AToday. And, regardless of the value of the parameter c, P(H) = P_(H|AToday) = 1/3. 5. Bayesian thirdism v. non-Bayesian thirdism As I said, it is very commonly assumed in the literature on the original Sleeping Beauty problem that Beauty acquires no new relevant informa- tion upon being awakened by the experimenters. Some thirders hold this view – notably Elga, who is explicit about it. These non-Bayesian thirders cannot and do not argue, as I do, that the basis for the change in P(H) from 1/2 on Sunday to 1/3 on Monday is Bayesian updating on newly acquired information. Nor can they reply to White, as I do, that White’s H-variability inference in the generalized Sleeping Beauty problem – and likewise his appeal to diachronic Bayesian updating via P_(H|A1) – are fallacious by virtue of ignoring relevant new information. On the assumption that Beauty gains no new relevant information in the original problem, and on the correlative assumption that the strongest new relevant information she gains in the generalized problem is A1, White’s H-variability inference does look very plausible – as does diach- ronic updating by means of P(H|A1). The generalized Sleeping Beauty problem thus poses a serious challenge to non-Bayesian thirders, even though it makes no trouble for Bayesian thirdism. So, unless and until the non-Bayesian thirders provide a plausible treatment of their own of the generalized problem, and a plausible account of their own of how White’s reasoning goes wrong, the capacity of Bayesian thirdism to smoothly handle the generalized problem provides new dialectical support for my own contention, over against the non-Bayesian thirders, that the right form of thirdism is Bayesian. The correct way to reason, in both the original and the generalized versions of the problem, is to invoke synchro- nic Bayesian updating on newly acquired indexical information.6 6 My thanks to Nathan Ballantyne, David Chalmers, Adam Elga, Justin Fisher, Michaela Mueller, Joel Pust, Mark Timmons and Roger White for helpful comments and discussion. quantum sleeping beauty 59 The University of Arizona Tucson, AZ 85721-0027, USA thorgan@email.arizona.edu References Arntzenius, F. 2003. Some problems for conditionalization and reflection. Journal of Philosophy 100: 356–70. Bostrum, N. In press. Sleeping Beauty and self-location: a hybrid model. Synthese. Dorr, C. 2002. Sleeping Beauty: in defence of Elga. Analysis 62: 292–96. Elga, A. 2000. Self-locating belief and the Sleeping Beauty problem. Analysis 60: 143–47. Horgan, T. 2004. Sleeping Beauty awakened: new odds at the dawn of the new day. Analysis 64: 10–21. Horgan, T. In press. Synchronic Bayesian updating and the Sleeping Beauty problem: reply to Pust. Synthese. Weintraub, R. 2004. Sleeping Beauty: a simple solution. Analysis 64: 8–10. White, R. 2006. The generalized Sleeping Beauty problem: a challenge for thirders. Analysis 66: 114–19. Blackwell Publishing Ltd.Oxford, UK and Malden, USAANALAnalysis0003-26382007 Blackwell Publishing Ltd.January 20076715965ArticlesPeter J. Lewis QUANTUM SLEEPING BEAUTY