Maximum A Posteriori Hypothesis

If too many H_i models, can't compute the sum. MAP hypothesis $H_{\rm MAP}$ is the one that maximizes $\Pr(H_i\vert D)$.If $\Pr(H_{\rm MAP}\vert D) \gt\gt \Pr(H_i\vert D)$, then

$$\Pr(X\vert D) \approx \Pr(X\vert H_{\rm MAP})\Pr(H_{\rm MAP}\vert D)$$

Why?

Note that (why?)

$$\Pr(H_i\vert D) = \frac{\Pr(D\vert H_i)\Pr(H_i)}{\Pr(D)}$$

so only need to consider $\Pr(D\vert H_i)\Pr(H_i)$ to pick $H_{\rm MAP}$.Why?

$\Pr(H_i)$ is the prior, uniform prior yields maximum likelihood hypothesis.