MaximumaPosterioriSchätzung

Maximum a Posteriori estimation, often abbreviated as MAP, is a method in Bayesian statistics for estimating an unknown parameter by maximizing the posterior distribution given observed data. The MAP estimate theta_MAP is defined as argmax_theta p(theta | D) = argmax_theta p(D | theta) p(theta) / p(D). Because p(D) does not depend on theta, this reduces to argmax_theta p(D | theta) p(theta). In practice, work with the log-posterior: theta_MAP = argmax_theta [ log p(D | theta) + log p(theta) ].

MAP combines likelihood information from the data with prior beliefs encoded in p(theta). A uniform prior yields the maximum likelihood estimate, while informative priors introduce regularization or prior knowledge. Conjugate priors can yield closed-form solutions; otherwise, numerical optimization is used on the log-posterior.

Common applications include machine learning models, signal processing, and computer vision. For instance, in regression or classification, MAP estimation with a Gaussian prior on parameters corresponds to L2 regularization (ridge). In Bernoulli problems with a Beta prior, the MAP estimate is (sum x + a - 1) / (n + a + b - 2).

MAP estimation contrasts with Maximum Likelihood Estimation (MLE) by incorporating prior information; it is a Bayesian estimator, not a frequentist one. When the prior is very informative, MAP can dominate the estimate; with weak priors results approach MLE.

Computational approaches include convex optimization for log-concave posteriors, gradient-based methods, Laplace approximations, variational inference, and Markov chain Monte Carlo for complex posteriors.