The Maximum-a-posteriori Estimator

The posterior mode loss function is

$$L(\theta ,\hat{\theta })=\{\begin{array}{ll}0,& \text{for }|\theta -\hat{\theta }|<\delta \\ 1,& \text{for }|\theta -\hat{\theta }|\ge \delta \end{array}$$

And minimizing the Bayes Risk (finding $\hat{\theta }$ ) gives us

$${\hat{\theta }}_{MAP}={\text{argmax}}_{\theta }p(x|\theta )p(\theta )$$

Which is very similar to The Maximum Likelihood Estimator, but with an added prior term.