Maximum a Posteriori Estimation

statistics machine-learning optimisation

Definition

Maximum a Posteriori Estimation

Maximum a Posteriori (MAP) estimation is a method for parameter estimation that incorporates prior beliefs about the distribution of parameters. Formally, given observed data $D$ and a prior distribution $P(\theta )$, the MAP estimator identifies the mode of the posterior distribution $P(\theta |D)$:

${\theta }_{\text{MAP}}^{\ast }=\arg {\max }_{\theta \in \Theta }P(\theta |D)=\arg {\max }_{\theta \in \Theta }P(D|\theta )P(\theta )$

Advantages

Regularisation: By incorporating $P(\theta )$, MAP acts as a natural form of regularisation, preventing the extreme parameter estimates that can occur in MLE when data is scarce.

Information Integration: It provides a formal mechanism to update initial beliefs (the prior) with empirical evidence (the likelihood) to reach a refined conclusion (the posterior).