Maximum a posteriori estimation

Maximum a posteriori estimation selects the parameter value with the highest posterior probability given observed data and a prior distribution.

Bayes' theorem: A rule relating conditional probabilities that updates prior probabilities using observed evidence. It supplies the probability update whose posterior MAP estimation maximizes.
Bayesian inference: Inference that represents uncertainty with probability distributions and updates them using evidence. MAP is one point-estimation summary within this broader inferential framework.
Regularization: Methods that constrain or penalize model fitting to reduce overfitting or improve stability. Many penalties correspond to negative log-priors in a MAP objective.
Maximum likelihood estimation: An estimation method that chooses parameters maximizing the likelihood of observed data. Unlike MAP, it does not incorporate a prior distribution.
Likelihood function: A function of model parameters that measures how well each parameter value explains observed data. Its peaks favor parameters that make the observed data more probable.
Statistical parameter: A quantity that specifies a model's probability distribution or other statistical behavior. MAP estimation seeks a value for an unknown model parameter.
Ridge regression: Linear regression with a squared-coefficient penalty that shrinks estimates toward zero. Its penalty yields a MAP interpretation under a zero-centered Gaussian prior.
Posterior mean: The expectation of an unknown quantity under its posterior distribution. It summarizes the posterior by an average, which need not be its mode.
Prior probability: A probability distribution expressing uncertainty about a quantity before incorporating specified evidence. Its influence can shift the posterior maximum away from the likelihood maximum.
Probability density function: A function whose integral over a region gives the probability of a continuous random variable lying there. For continuous parameters, MAP commonly maximizes posterior density rather than point probability.

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