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In general there will be many situations producing a given set of detector readings; let  be the set of situations in  producing the particular n-tuple of readings (v1, . . ., vn). In probabilistic terms, is an event defined on the sample space . Events themselves can be treated as random variables. (In fact, an occurrence of the situation can be construed as the occurrence of all the events of which it is an instance.) Moreover, the function W assigning values to elements of can be restricted to the event so that it becomes a random variable W(v1, . . . , vn) over . As such W(v1, . . . , vn) has a well-defined expected value  over . |
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This probabilistic view of search plans is closely related to statistical inference based on sampling plans. The estimation of from observation of a few samples drawn from is a standard problem of statistical inference. We can think of a subset of detectors H as detecting one kind of critical feature when the corresponding is greater than  , where is the average value of the random variable W over the sample space . Search plans go further in attempting to infer something of the value of for which have not been sampled. For example,  is contained in both  and  ; often it is possible to infer something of  from knowledge of  and  , though not necessarily by standard statistical techniques. |
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The earlier concern with distinguishability is also directly stated in these terms: Let d(t) be the particular n-tuple of detector readings at time t  and let  be a search plan. That is, f is a prescription which specifies, for each set of detector readings, a transformation. The object of the search plan is to transform the current situation into one of high utility. But, for this to be possible, the effects of the transformations must be reliably indicated by the detectors. In particular, consider S1 and  , so that at t = 1 either would show the same reading  The plan f specifies the action h(1) = f(d(1)), and this in turn produces a new detector reading d(2). The whole procedure is iterated to yield a sequence of pairs   . The requirement on distinguishability is simply that, using the information provided by the detectors, f reliably transforms S1 and S2 into situations  and  , respectively, for which  . (Notice that this is a much weaker requirement than would be necessary for a completely "autonomous" model wherein future situations would be wholly predictable on the basis of d(1) without any further information from the environment. That is, in an autonomous model, knowledge of d(1) and h(1), . . , h(t) must suffice to determine d(t + 1). This requirement for "autonomy"technically a requirement that the detectors induce a homomorphismcan be quite difficult to meet and, for intricate |
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