The predictive distribution in decision theory: a case study

In the classical decision theory framework, the loss is a function of the decision taken and the state of nature as represented by a parameter 0. Information about 0 can be obtained via observation of a random variable X. In some situations however the loss will depend not directly on 0 but on the observed value of another random variable Y whose distribution depends on 0. This adds an extra layer to the decision problem, and may lead to a wider choice of actions. In particular there are now two sample sizes to choose, for X and for Y, leading to a range of behaviours in the Bayes risk. We illustrate this with a problem arising from the cleanup of sites contaminated with radioactive waste. We also discuss some computational approaches.