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 θ. Information about θ can be obtained via observation of a random variable X. In some situations however the loss will depend not directly on θ but on the observed value of another random variable Y whose distribution depends on θ. 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 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.