Maximum a posteriori estimation extends maximum likelihood by multiplying the likelihood by a prior distribution over the parameters and maximizing the product; because the marginal probability of the data does not depend on the parameter, the normalizing denominator can be dropped, so the posterior itself never has to be computed and the objective is conveniently optimized in log form. The prior must be supported on the parameter's admissible range — for a probability parameter this rules out an unbounded distribution and motivates the beta distribution, whose two shape parameters act as pseudo-counts added to the observed successes and failures. The method belongs to Bayesian statistical estimation within machine learning, and it supplies a probabilistic reinterpretation of regularization: penalties on parameter magnitude or on the number of nonzero parameters correspond to priors that assign low probability to large or dense solutions.
Maximum a posteriori estimation extends maximum likelihood by multiplying the likelihood by a prior distribution over the parameters and maximizing the product; because the marginal probability of th…