Bayesian MSE Estimators

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The Bayesian Mean Square Error of the estimator is

$$\begin{array}{rl}\text{Bmse}(\hat{\theta })& =E[(\theta -\hat{\theta })]\\ & =\int \int (\theta -\hat{\theta }{)}^{2}p(x,\theta )d\theta dx\end{array}$$

Finding $\hat{\theta }$ by solving $\hat{\theta }={\text{argmin}}_{{\theta }^{\prime }}E[(\theta -\hat{\theta })]$, and taking the derivative wrt. $\hat{\theta }$ to zero yields

$$\hat{\theta }=\int \theta p(\theta |x)d\theta =E[\theta |x]$$

The Bayesian MSE estimator is the conditional expectation of the paramter $\theta$ given our observations $x.$