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 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.