MAP Estimation

Overview

Maximum A Posteriori (MAP) estimation finds the mode of the posterior distribution:

\[ \hat{\theta}_{MAP} = \arg\max_{\theta} \pi(\theta \mid \mathbf{x}) = \arg\max_{\theta} [\log f(\mathbf{x} \mid \theta) + \log \pi(\theta)] \]

Relationship to MLE

MAP estimation differs from MLE by adding a log-prior term. As \(n \to \infty\), MAP and MLE converge because the likelihood dominates the prior.

Relationship to Regularization

• Gaussian prior \(\to\) L2 (Ridge) regularization
• Laplace prior \(\to\) L1 (Lasso) regularization