The predictive distribution in decision theory: A case study (Q1281338)

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