The maximum a posteriori estimator (MAP) is a single value representation of the [[posterior]] distribution, analogous to the [[maximum likelihood estimator]] for [[frequentist]] methods. It is the value that maximizes the posterior distribution function. $ \hat \theta = \arg \max \pi(\theta | x) $ As with the MLE, the $\arg \max$ of the log-likelihood of the posterior will be equivalent to that of the likelihood. Using [[logarithm rules]], another way of specifying the MAP is $ \hat \theta = \arg \max \Big[ \ln f(x|\theta) + \ln \pi(\theta) \Big] $