Maximum a Posteriori Estimation

The Maximum a Posteriori estimator, or maximum a posteriori probability (MAP), provides one of the modes of the posterior distribution as its estimate. Unlike maximum-likelihood estimation, MAP estimation obtains a posterior density using Bayesian theory by combining the prior knowledge $f(\boldsymbol\vartheta)$ with the conditional likelihood density $\mathcal{L}(\boldsymbol\vartheta \vert \mathbf{x}) = f(\mathbf{x} \vert \boldsymbol\vartheta)$, yielding the new estimate

$$\hat{\boldsymbol\vartheta}_{MAP} = \argmax_{\boldsymbol\vart... ...mathbf{x} \vert \boldsymbol\vartheta) f(\boldsymbol\vartheta)$$ (2.61)

and, in the case of uncorrelated events, the formula becomes

$$\hat{\boldsymbol\vartheta}_{MAP} = \argmax_{\boldsymbol\vart... ...\boldsymbol\vartheta) \right\} + \log f(\boldsymbol\vartheta)$$ (2.62)

where, again for computational convenience, the properties of the logarithm have been used.

Clearly, if the prior probability $f(\boldsymbol\vartheta)$ is uniform, MAP and MLE coincide.