Maximum A Posteriori

Prior: the initialized parameter for the assumed distribution of parameter

$$\begin{array}{rl}p(\theta |D)& =\frac{p(D|\theta )p(\theta )}{p(D)}\\ & =\frac{\prod p(x|\theta )p(\theta )}{\int p(D|\theta )p(\theta )d\theta }\end{array}$$

Given a dataset, adjust the distribution of parameter. Find the parameter that maximizes the probability of the parameter. It can be written as:

\begin{align} \hat \theta\_{MAP} (x) =& \operatorname\*{argmax}_\theta p(\theta | x) \\ \=& \operatorname\*{argmax}_\theta \frac{f(x | \theta) g(\theta)}{\int f(x | \theta') g(\theta') d\theta'} \\ \=& \argmax\_\theta f(x|\theta)g(\theta) \end{align}