  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.
