Anthropic Bias Observation Selection Effects in Science and Philosophy Nick Bostrom
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- How the reference class may be observer-moment relative
Reassessing IncubatorNext we home in on a key lesson that emerges from the preceding investigations: that the critical point, the fountainhead, of all the paradoxical results seems to be the contexts where the hypotheses under consideration have different implications about the total number of observers in existence. Such is the way with DA, the various Adam & Eve experiments, Quantum Joe, and UN++. By contrast, things seem to be humming along perfectly nicely so long as the total number of observers is held constant. Here is another clue: Recall that we remarked in chapter 4 that the cases where the definition of the reference class is relevant for our probability assignments seem to be precisely those where the total number of observers depends on which hypothesis is true. This suggests that the solution we are trying to find has something to with how the reference class is defined. So that we may focus our beam of attention as sharply as possible on the critical point, let us contemplate the simplest case where the number of observers is a variable and that we can use to model the reasoning in DA and the problematic thought experiments: Incubator. Now that we have SSSA, it is useful to add some details to the original version:
The incubator tosses a fair coin in an otherwise empty world. If the coin falls heads, the incubator creates one room with a black-bearded observer and one room with one white-bearded observer; if it falls tails, the incubator creates only a room with a black-bearded observer. Any observers who have been created first spend one hour in darkness (when they are ignorant about their beard color), and then one hour with the lights on (so they can see their beard in a mirror). Everyone knows the setup. The experiment ends after two hours whereupon everybody is killed. The situation is depicted in figure 5. For simplicity, we can assume that there is one observer-moment per hour per observer.  We discussed three models for how to reason about Incubator in chapter 4. We rejected Model 1 and Model 3 and were thus left with Model 2 – the model which we have subsequently seen leads to counterintuitive results. Is it perhaps possible that there is fourth model, a better way of reasoning that can be accessed by means of the more powerful analytical resources provided by SSSA? Let’s consider again what the observer-moments in Incubator should believe. To start with, suppose that all observer-moments are in the same reference class. Then it follows from directly from SSSA that84 P(“This is an observer-moment that knows it has a black beard” | Tails) = 1/2 P(“This is an observer-moment that knows it has a black beard” | Heads) = 1/4 Together with the fact that the coin toss is known to have been fair, this implies that after the light comes on, the observer-moment who knows it has a black beard should assign a credence of 1/3 to Heads. This is the conclusion that, when transposed to DA and the Adam & Eve thought experiments, leads to the problematic probability shift in favor of hypotheses that imply fewer additional observers.85 This suggests that if we are unwilling to accept these consequences, we should not place all observer-moments in the same reference class. Suppose that we instead put the early observer-moments in one reference class and the late observer-moments in separate reference classes. We’ll see how this move might be justified in the next section, but we can already note that making the choice of reference class context-dependent in this way is not entirely arbitrary. The early observer-moments, which are in very similar states, are in the same reference class. The observer-moment that has discovered that it has a black beard is in an importantly different state (no longer wondering about its beard color) and is thus placed in a different reference class. The observer-moment that has discovered it has a white beard is again different from all the other observer-moments (it is, for instance, in a state of no uncertainty as to its beard color and can deduce logically that the coin fell heads), and so it also has its own reference class. The differences between the observer-moments are significant at least in the respect that they concern what information the observer-moments have that is relevant to the problem at hand, viz. to guess how the coin fell. If we use this reference class partitioning then SSSA no longer entails that the observer-moment who has discovered that it has a black beard should favor the Tails hypothesis. Instead, that observer-moment will now assign equal credence to either outcome of the coin toss. This is because on either Tails or Heads, all observer-moments in its reference class (which is now the singleton consisting only of that observer-moment itself) observe what it is observing; so SSSA gives: P(“This is an observer-moment that knows it has a black beard” | Tails) = 1 P(“This is an observer-moment that knows it has a black beard” | Heads) = 1 The problematic probability shift is thus avoided. It is remains the case that the early observer-moments, who are ignorant about their beard-color, assign an even credence to Heads and Tails; so we have not imported the illicit SIA criticized in chapter 7. As for the observer-moment that discovers that it has a white beard, SSSA gives the following conditional probabilities: P(“This is an observer-moment that knows it has a black beard” | Tails) = 0 P(“This is an observer-moment that knows it has a black beard” | Heads) = 1 So that observer-moment is advised to assign zero credence to the Tails hypothesis (which would have made its existence impossible). How the reference class may be observer-moment relativeCan it be permissible for different observer-moments to use different reference classes? We can turn this question around by asking: Why should different observer-moments not use different reference classes? What argument is there to show that such a way of assigning credence would necessarily be irrational? In chapter 4, we gave an argument for accepting Model 2, the model asserting that the observer who knows he has a black beard should assign a greater than even credence to Tails. The argument had the following form: First consider what you should believe if you don’t know your beard color; second, in this state of ignorance, assign conditional probabilities to you having a given beard color given Heads or given Tails; third, upon learning your beard color, use Bayesian kinematics to update the credence function obtained through the first two steps. The upshot of this process is that after finding that you have a black beard your credence of Tails should be 2/3. Let’s try to recapture this chain of reasoning in our present framework using observer-moments. The early observer-moments don’t know whether they have black or white beard, but they can consider the conditional probabilities of that given a particular outcome of the coin toss. They know that on Heads, one out of two of the observer-moments in their epistemic situation has a black beard; and on Tails, one out of one has a black beard: P(“This observer-moment has a black beard” | Tails & Early) = 1 P(“This observer-moment has a black beard” | Heads & Early) = 1/2 (“Early” stands for “This observer-moment exist during the first hour”.) One can easily see that this credence assignment is independent of whether the universal reference class is used or one uses the partition of reference classes described above. Moreover, since the observer-moments know that the coin toss is fair, they also assign an even credence to Heads and Tails.86 This gives (via Bayes’ theorem): P(Tails | “This observer-moment has a black beard” & Early) = 2/3 (C1) P(Heads | “This observer-moment has a black beard” & Early) = 1/3 (C2) When the lights come on, one observer discovers he has a black beard. The argument under investigation would now have him update his credence by applying Bayesian conditionalization to the conditional credence assignments (C1 & C2) that he made when he was ignorant about his beard color. And this where the argument fails. For the later observer-moment’s evidence is not equivalent to the earlier observer-moment’s evidence conjoined with the proposition that it has a black beard. The later observer-moment has also lost knowledge of the indexical proposition “Early” and moreover, the indexical proposition expressed by “This observer-moment has a black beard” is a different one when the thought is entertained by the later observer-moment, since “this” then refers to a different observer-moment. Therefore, we see that the argument that would force the acceptance of Model 2 relies on the implicit premiss that the only relevant epistemological difference between the observer before and after he discovers his beard color is that he gains the information that is taken into account by the Bayesian conditionalization referred to in step three. If there are other relevant informational changes between the “early” and the “late” states of the observer, then there is no general reason to think that his credence assignments in the latter state should be obtained by simply conditionalizing on the finding that he is an observer with black beard. In chapter 4, were we had by stipulation limited our consideration to including only such indexical information as concerned which observer one is, this hidden premiss was satisfied; for the latter state of the observer then differed from the early one in precisely one regard, namely, by having acquired the indexical information that he is the observer with the black beard – the information that was conditionalized on in step three. Now, however, this tacit assumption is no longer supported. For we now have also to consider changes in other kinds of indexical information that might have occurred between the early and the late stages. This includes the change in the indexical information about which temporal part of the observer (i.e. which observer-moment) one currently is. Before the observer finds that he has a black beard, he knows the piece of indexical information that “this current observer-moment is one that is ignorant about its beard color”. After finding out that he has a black beard, he has lost that piece of indexical information (the indexical fact no longer obtains about him); and the information he has gained includes the indexical fact that “this current observer-moment is one that knows that it has a black beard”. These differences in information (which the argument for Model 2 fails to take into account) could potentially be relevant to what credence the observer should assign to the Tails and Heads hypotheses after he has found out that he has a black beard. Consider now the claim that the reference class is observer-moment relative, more specifically, that the early and the late observer-moments should use different reference classes, as described above. Then, since the reference class is what determines the conditional probabilities that are used in the calculation of the posterior probabilities of Heads or Tails, we have to acknowledge that the difference in indexical information just referred to is directly relevant and must therefore be taken into account. The indexical information that the early observer-moments use to derive the conditional probabilities C1 and C2 (namely, the indexical information that they are early observer-moments, which is what determines that their reference class is, which in turn determines these conditional probabilities) is lost and replaced by different indexical information when we turn to the later observer-moments. The later observer-moments, having different indexical information, belong, ex hypothesi, in a different reference classes mandating a different set of conditional probabilities. If a late observer-moment’s reference class does not include early observer-moments, then its conditional probability (given either Heads or Tails) of being an early observer-moment is zero. Conditionalizing on being a late observer-moment would therefore have no influence on the credence that the late observer-moment assigns to the possible outcomes of the coin toss. (The late observer-moment that has discovered it has a white beard has of course got another piece of relevant information, which implies Tails, so that’s what it should believe, with probability unity.) The argument I’ve just given does not show that the difference in indexical information about which observer-moment one currently is requires that different reference classes be used. All it does is to show that this is now an open possibility, and that the argument to the contrary that was earlier used to support model 2 can no longer be applied once the purview is expanded to SSSA which takes into account a more complete set of indexical information. What this means is that the arguments relying on Model 2 can now be seen to be inconclusive; they don’t prove what they set out to prove. We are therefore free to reject DA and the assertion that Adam and Eve, Quantum Joe and UN++ should believe the counterintuitive propositions which, if the sole basis of evaluation were the indexical information taken into account by SSA, they would have been rationally required to accept. Indeed, the fact that the choice of a universal reference class leads to the implausible conclusions of chapter 9 is a reason for rejecting the universal reference class as the exclusively rational alternative; it suggests that, instead, choosing reference in a more context-dependent manner is a preferable method. I am not claiming that this reason is conclusive. One could choose to accept the consequences discussed in Adam & Eve, Quantum Joe and UN++. If one is willing do that then nothing that has been said here stops one from using a universal reference class. But if one is unwilling to embrace those results then the way in which one can coherently avoid doing so is by insisting that one’s choice of reference class is to some degree dependent on context (specifically, on indexical information concerning which observer-moment one currently is). The task now awaiting us is to explain how an observation theory can be developed that meets all the criteria and desiderata listed above that that can operate with a relativized reference class. The theory we shall propose is neutral in regard to the reference class definition. It can therefore be used either with a universal reference class or with a relativized reference class; the theory specifies how credence assignments are to be made given a choice of reference class. This is a virtue because in the absence of solid grounds for claiming that only one particular reference class definition can be rationally permissible, it would be wrong to rule out other definitions by fiat. This is not to espouse a policy of complete laissez-faire as regards the choice of reference class. We shall see that there are interesting limits on the range of permissible choices. 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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