Anthropic Bias Observation Selection Effects in Science and Philosophy Nick Bostrom
Chapter 12: The Sleeping-Beauty problem: modelling imperfect recall
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Chapter 12: The Sleeping-Beauty problem: modelling imperfect recallThe Sleeping Beauty ProblemOn Sunday afternoon Beauty is given the following information: She will be put to sleep on Sunday evening and will wake up on Monday morning. Initially she will not be told what day it is, but on Monday afternoon she’ll be told it is Monday. Then on Monday evening she will be put to sleep again. Then a fair coin is tossed and if and only if it falls tails will she be awakened again on Tuesday. However, before she is woken she will have her memory erased so that upon awakening on Tuesday morning she has no memory of having been awakened on Monday. When she wakes up on Monday, what probability should she assign to the hypothesis that the coin landed heads? Sleeping Beauty is a problem involving imperfect recall. Two other such problems that have been discussed in recent years in the game theory literature are the Absent-Minded Driver and the Absent-Minded Passenger. The three being closely linked, one’s view on one of them is likely to determine what one thinks about the others. Therefore we can regard Sleeping Beauty as a template for dealing with a broader class of imperfect recall problems. In Sleeping Beauty, views diverge as to whether the correct answer is P(Heads) = 1/3 or P(Heads) = 1/2. In support of the former alternative is the consideration that if there were a long series of Sleeping Beauty experiments then on average one third of the awakenings would be Heads-awakenings. One might therefore think that on any particular awakening, Beauty should believe with a credence of 1/3 that the coin landed heads in that trial. In support of the view that P(Heads) = 1/2 there is the consideration that the coin is known to be fair and it appears as if awakening does not give relevant new information to somebody who knew all along that she would at some point be awakened. The former view is advocated in e.g. ((Elga 2000)) and the latter in e.g. ((Lewis 2001)); but see also ((Aumann, Hart et al. 1997; Battigalli 1997; Gilboa 1997; Grove 1997; Halpern 1997; Lipman 1997; Piccione and Rubinstein 1997; Piccione and Rubinstein 1997; Wedd 2000)) for earlier treatments of the same or similar problems. My position is that the issue is more complicated than existing analyses admit and that the solution is underdetermined by the problem as formulated above. It contains ambiguities that must be recognized and disentangled. Depending on how we do that, we get different answers. In particular, we need to decide whether there are any outsiders (i.e. observer-moments other than those belonging to Beauty while she is in the experiment), and what Beauty’s reference class is. Once these parameters have been fixed, it is straightforward to calculate the answer using OE. The case of no outsidersLet’s consider first the case of no outsiders. Thus, we can suppose that Beauty is the only observer in the world and that she is created specifically for the experiment and that she is killed as soon as it is over. We can simplify by representing each possible period of being awake as a single possible observer-moment. (As shown earlier, it makes no difference how many observer-moments we associate with a unit of subjective time, provided we use a sufficiently fine-grained metric to accurately represent the proportions of subjective time spent in the various states.) We can then represent Sleeping Beauty graphically as follows (figure 12):  The diagram shows the possible observer-moments, and groups those that have the same total information together in equivalence classes. Thus, for instance,  ,  , and  denote the “Monday-morning-in-the-Tails-world” observer-moment, the “Tuesday-morning-in-the-Tails-world” observer-moment, and the “Monday-morning-in-the-Heads-world” observer-moment, respectively. Since they have the same evidence (not shared by any other observer-moment), they constitute an equivalence class. This equivalence class, which we have denoted by “ ”, represents the evidence that each of these observer-moments has.
To find the solution, we must also know Beauty’s reference class. Consider first the case where she includes all observer-moments in her reference class: Directory: preprints preprints -> Digitally-assisted discovery and proof preprints -> Reorienting the Logic of Abduction preprints -> Supplementary Material Table 1 Download 9.31 Mb. Share with your friends:
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