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Combining the Gaussian likelihood with the prior $\theta \sim N_p(0,c^2{I}_p)$ gives the posterior precision and meanThus the posterior in the Gaussian normal-mean shrinkage Bayes estimator isThe Bayes estimator under squared error loss is the posterior mean, soIts posterior expected loss is the trace of the posterior covariance and does not depend on $X$. Therefore its Bayes risk is