Bayes risk

Remarks

The integral is a Lebesgue integral of a nonnegative function, so it always exists, possibly infinite. By Fubini's theorem, $r(\pi, \delta)$ is also the expected loss when $\theta$ is drawn from $\pi$ and then $X$ from $P_\theta$.

Example: let $X$ be binomial with parameters $n$ and $\theta \in [0, 1]$, with squared error loss and $\delta(X) = X/n$. Then $R(\theta, \delta) = \theta(1 - \theta)/n$. For the uniform prior, $r(\pi, \delta) = \frac{1}{n} \int_0^1 \theta(1 - \theta) \, d\theta = \frac{1}{6n}$.

Non-example: the maximum $\sup_\theta R(\theta, \delta) = \frac{1}{4n}$ is not a Bayes risk of $\delta$ for the uniform prior. It is the worst-case risk used in the minimax principle; it equals $r(\pi, \delta)$ only for priors that put all their mass where the risk is maximal.