03 Maximum A Posteriori Estimation

• Maximum A Posteriori (MAP) : The most probable value of $\theta$ is where the posterior takes it maximum

$$\begin{array}{rl}{\theta }_{\text{MAP}}& =\arg \underset{\theta }{\max }p(\theta |D)\\ & =\arg \underset{\theta }{\max }\left\{\log [p(D|\theta )]+\log [p(\theta )]\right\}\end{array}$$

• MAP use optimisation to find the parameter value that maximises the posterior probability.