Maksimiposteriori

Maksimiposteriori, or maximum a posteriori estimation, is a method in Bayesian statistics to estimate an unknown parameter by selecting the value that maximizes the posterior distribution given observed data. The posterior is proportional to the likelihood times the prior: p(theta|D) ∝ p(D|theta) p(theta). The MAP estimate is theta_MAP = argmax_theta p(D|theta) p(theta), equivalently theta_MAP = argmax_theta [log p(D|theta) + log p(theta)]. If the prior p(theta) is uniform (or very flat), the MAP estimate reduces to the maximum likelihood estimate (MLE).

The MAP approach uses prior information to regularize estimates, which can be beneficial in small-sample situations or when the parameter space is ill-posed. However it introduces sensitivity to the chosen prior, particularly when data are limited. In many conjugate models, the MAP estimate has a closed-form solution.

In practice, computing the MAP estimate often requires optimization techniques. The MAP is one point estimate; by contrast, the full Bayesian approach maintains the posterior distribution, from which the posterior mean or other summaries can be derived.

Common contexts for MAP include parameter estimation in statistics, machine learning, signal processing, and pattern recognition. The term Maksimiposteriori reflects the Finnish or general usage of the concept, but is widely understood in many languages that use similar abbreviations (MAP).