On Bayes risk of the posterior mean in linear inverse problems

We consider the concept of Bayes risk in the context of finite-dimensional ill-posed linear inverse problem with Gaussian prior and noise models. In this note, we rederive the following well-known result: in the present Gaussian linear setting, the Bayes risk of the posterior mean, relative to the sum of squares loss function, equals the trace of the posterior covariance operator. While our discussion is limited to a finite-dimensional setup, our presentation is motivated by more general infinite-dimensional formulations.