Bayes risk

For a prior $\pi$, the integrated risk is $r(\pi ,\delta )=\int R(\theta ,\delta )\pi (d\theta )$. Its infimum over decision rules is the Bayes risk, attained by a Bayes rule when one exists.

A Bayes act minimizes $\int L(\theta ,a)\Pi (d\theta )$ for a specified parameter distribution $\Pi$. Applying the same minimization to the posterior after observing data gives a Bayes decision rule. The optimal data-based Bayes risk is the expectation of the optimal posterior no-data loss.

For a probability density $\pi$ and threshold $c>0$, this region includes precisely the points at or above the threshold. It is a Bayes act for loss $c|S|-1_S(\theta )$, since expected loss is ${\int }_S(c-\pi (\theta ))d\theta$. The minimal value is $-\int (\pi -c{)}_{+}d\theta$; adding or removing threshold-equality points does not change it.

A least favorable prior maximizes the Bayes risk over the allowed priors. If a Bayes rule has constant risk equal to the minimax value, its prior is least favorable.