C7. MAP and squared-error loss

Topic: Bayesian foundations · more questions on this topic
‹ C6

How does a MAP estimate differ from a Bayes estimate under squared-error loss?

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Definitions/notation. The MAP maximises p(\theta\mid x), whereas a Bayes estimate minimises the posterior expected loss for a specified loss function. Under squared-error loss and a finite posterior second moment, \mathbb{E}[(\theta-d)^2\mid x]=\operatorname{Var}(\theta\mid x)+(\mathbb{E}[\theta\mid x]-d)^2, so the Bayes estimate is the posterior mean, not generally the mode.