By D V Lindley
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The value of experiments. 7) we saw how to solve the experimental design problem within the Bayesian framework. 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.
An example of the use of these ideas in a bioassay situation is given by Freeman (1970). His problem is to estimate the relationship, supposed parameterized in terms of 9, between Z, the percentage of animals affected by a drug when applied at a dosage of strength e (the notation is chosen to fit with the present theory). For the ith animal the dosage et may be selected, the response being x,- = 1 or 0 according to whether or not the animal is affected. A famous example is that of the two-armed bandit.
Consequently in many cases it is possible to derive a posterior distribution by considering only the likelihood function and certain modest properties of the prior. They only discuss the case where 0 is the real line. Additional difficulties can arise when 0 is of higher dimension (examples will occur later) but clearly the same general principles obtain. 10) since the posterior is an easily calculable member of the same family. We illustrate with the case of binomial sampling which has received much attention.