Maximum a Posteriori Estimation

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

$$\hat{\boldsymbol\vartheta}_{MAP} = \argmax_{\boldsymbol\vart... ...mathbf{x} \vert \boldsymbol\vartheta) f(\boldsymbol\vartheta)$$ (2.58)

and in the case of uncorrelated events, the formula transforms into

$$\hat{\boldsymbol\vartheta}_{MAP} = \argmax_{\boldsymbol\vart... ...\boldsymbol\vartheta) \right\} + \log f(\boldsymbol\vartheta)$$ (2.59)

where, to simplify the calculations, the properties of logarithms have been utilized.

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