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A Bayes estimator chooses an action minimizing posterior expected loss after observing data:The answer depends on the loss function: squared error gives the posterior mean, absolute error a posterior median, and discrete zero-one loss a posterior mode. For squared error, expanding $\mathbb{E}[(\theta -a{)}^2\mid x]=\mathrm{Var}(\theta \mid x)+(a-\mathbb{E}[\theta \mid x]{)}^2$ proves the mean rule when the second moment is finite. Pointwise posterior minimization also minimizes integrated Bayes risk under the usual measurability and integrability conditions.