 Posterior expected loss

Given observed data $x$, the posterior expected loss of action $a$ iswhere $\Pi (\cdot \mid x)$ is the posterior distribution and $L$ the loss function. Nonnegative losses define an extended nonnegative integral; meaningful comparisons require checking when finite loss is attainable. A Bayes decision rule chooses a measurable minimizer, or an approximate minimizer if an optimum is not attained.

With finite posterior second moment and squared-error loss, completing the square gives $\rho (a\mid x)=\mathrm{Var}(\Theta \mid x)+(a-\mathbb{E}[\Theta \mid x]{)}^2$. Thus the posterior mean minimizes the loss. With finite first moment and absolute-error loss, a posterior median minimizes it instead. Posterior expected loss conditions on the observed data and averages over parameter uncertainty; the frequentist risk of a decision rule fixes the parameter and averages over possible data.