Bayesian statistics, a review by D V Lindley

By D V Lindley

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This expression is now studied further in order to assess the value of an experiment e. We suppose U(d, 9, e, x) = U(d, 9) + U(x, e) so that the terminal utility and experimental costs are additive. The expected utility of e before it is performed is Consider the second of the two terms in the braces. It equals the expected utility of the best decision from e, given that x is observed. Hence the expectation of the utility from e will be the average of this over X. Whereas if e is not performed the best that can be obtained is maxd U(d, 9)p(9) d9.

According to the Bayesian canon all uncertain quantities are specified by probabilities, so that here there exists p(0), the distribution of 0 (the population description) prior to sampling. Then the sampling rule provides a conditional density p(s|0). For example, if the rule is to sample randomly without replacement for all s. We saw that within the Bayesian framework randomization is unnecessary; if this is avoided, then p(s|0) is degenerate, being 1 for the selected s and otherwise zero. Typically p(s|0) does not depend on 0 but sometimes it does as when sampling fibres, the chance of a fibre being included in s depending on its unknown length.

Here, in formal language, each £, contains two elements, each of which gives a binomial trial of unknown chance 9{ in one case and 92 in the other. The losses are expressed naturally in terms of failures on the trials. The topic has an extensive literature (see, for example, De Groot (1970)). 1) provides a completely general method of solving the problems of sequential experimentation, in practice the analysis is involved and even the computation of numerical solutions is typically prohibitive.

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