On Bayes risk of the posterior mean in linear inverse problems
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We recall some basics regarding the concept of Bayes risk in the context of finite-dimensional ill-posed linear inverse problem with Gaussian prior and noise models. In particular, we rederive the following basic 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.
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