Bayes estimator

A Bayes estimator minimizes posterior expected loss.

A Bayes decision rule maximizes posterior expected utility, equivalently minimizes posterior expected loss, separately at each observed data value.

Under squared error loss, the Bayes estimator is the posterior mean.

For $X\mid \theta \sim N_p(\theta ,I_p)$ and $\theta \sim N_p(0,c^2{I}_p)$, the posterior isThe Bayes estimator under quadratic loss is the posterior mean and its Bayes risk is the trace of the posterior covariance, $p{c}^2/(1+c^2)$.

For loss $L(\theta ,a)=q(\theta )(\theta -a{)}^2$ with $q(\theta )>0$, the Bayes estimator is the mean under the posterior reweighted by $q$:

Under $L(d,\theta )={\theta }^{-1}(\theta -d{)}^2$, the posterior expected loss is minimized atprovided the conditional inverse moment is finite and positive.

Posterior expected loss averages the loss over the posterior and is minimized pointwise in the data.