General Bayesian Estimators

We define a loss function $L(\theta ,\hat{\theta })$

• Squared error: $L(\theta ,\hat{\theta })=(\theta -\hat{\theta }{)}^2$.
  • Discussed in Bayesian MSE Estimators
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• Absolute error: $L(\theta ,\hat{\theta })=|\theta -\hat{\theta }|$.
  • An alternative to squared error which does not penalize large errors too much
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• Posterior mode: $L(\theta ,\hat{\theta })=\{\begin{array}{ll}0,& \text{for }|\theta -\hat{\theta }|<\delta \\ 1,& \text{for }|\theta -\hat{\theta }|\ge \delta \end{array}$ .
  • Also called “hit-or-miss error”
  • The Maximum-a-posteriori Estimator
  • Any error outside a constant $\delta$ is penalized
  • We want to minimize the Bayes Risk, aka minimize

$$\int L(\theta ,\hat{\theta })p(\theta |x)d\theta$$

with respect to $\hat{\theta }\forall x$ to find the optimal estimator.