By Marcus R.

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**Extra resources for A Bayesian procedure for the sequential estimation of the mean of a negative-binomial distribution**

**Sample text**

To verify this, we seek rst a formula for the logical sum A + B . Applying the product rule and sum rule repeatedly, we have 210 2: Qualitative Properties 210 p(A + BjC ) = 1 and nally, p(A B jC ) = 1 p(AjC ) p(BjAC ) = 1 p(AjC )[1 p(B jAC )] = p(AjC ) + p(AB jC ) = p(AjC ) + p(B jC ) p(AjBC ) = p(AjC ) + p(B jC )[1 p(AjBC )] p(A + B jC ) = p(AjC ) + p(BjC ) p(ABjC ) : (2{48) This generalized sum rule is one of the most useful in applications. Evidently, the primitive sum rule (2{47) is a special case of (2{48), with the choice B = A.

The background information B stipulates that one and only one of them must be true. In that case the sum (2{64) for m = n must be unity: n X i=1 p(AijB) = 1 : (2{65) This alone is not enough to determine the individual numerical values p(Ai jB ). Depending on further details of the information B , many dierent choices might be appropriate, and in general nding the p(AijB ) by logical analysis of B can be a dicult problem. It is, in fact, an open{ended problem, since there is no end to the variety of complicated information that might be contained in B ; and therefore no end to the complicated mathematical problems of translating that information into numerical values of p(Ai jB ).

In particular, common language, being in constant use for other purposes than logic, has developed subtle nuances { means of implying something without actually stating it { that are lost Indeed, some psychologists think that as few as ve dimensions might suce to characterize a human personality; that is that we all dier only in having dierent mixes of ve basic personality traits which may be genetically determined. But it seems to us that this must be grossly oversimplied; identiable chemical factors continuously varying in both space and time (such as the distribution of glucose metabolism in the brain) aect mental activity but cannot be represented faithfully in a space of only ve dimensions.