Finite posterior risk and Bayes action rule

Let $(\mathrm{\Theta },\mathcal{T})$ have a supplied posterior probability kernel ${\mathrm{\Pi }}_y$, let $(\mathsf{A},\mathcal{G})$ be a measurable action space, and let $L:\mathrm{\Theta }\times \mathsf{A}\to [0,\mathrm{\infty }]$ be a measurable function on the product measurable space The product sigma-algebra and its finite iterates, in the sense of A measurable function between measurable spaces. This strengthens the section-measurability convention in Loss functions and risk so that posterior integrals vary measurably with measurable action rules.

For a fixed data fiber $y$ and action $a$, define the posterior risk only when ${\rho }_y(a):={\int }_{\mathrm{\Theta }}L(\theta ,a){\mathrm{\Pi }}_y(d\theta )<\mathrm{\infty }\mathrm{.}$ A pointwise Bayes action at $y$ is a finite-risk action attaining the minimum among all finite-risk actions. On a declared measurable data domain $D$, a Bayes action rule is a measurable map $\delta :D\to \mathsf{A}$ whose finite posterior risk attains that minimum at every $y\in D$. This definition makes no assertion that finite-risk actions or measurable minimizing rules always exist. It therefore does not infer a Bayes action merely from the existence of an infimum. The sampling risk in Loss functions and risk is a different integral, taken over the data law at a fixed parameter.