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:

$$\begin{array}{rl}{\hat{\theta }}_{MAP}(x)=& \underset{\theta }{\mathrm{argmax}}p(\theta |x)\\ =& \underset{\theta }{\mathrm{argmax}}\frac{f(x|\theta )g(\theta )}{\int f(x|{\theta }^{\prime })g({\theta }^{\prime })d{\theta }^{\prime }}\\ =& \underset{\theta }{\mathrm{argmax}}f(x|\theta )g(\theta )\end{array}$$