Concept - Maximum A Posteriori

Definition (s.18): find the most probable hypothesis given the training data (MAP hypothesis $h_{MAP}$). $h_{MAP}=\arg {\max }_{h\in \mathcal{H}}P(h|D)=\arg {\max }_{h\in \mathcal{H}}\frac{P(D|h)P(h)}{P(D)}=\arg {\max }_{h\in \mathcal{H}}P(D|h)P(h)$ $P(D)$ does not depend on $h$ so it is dropped. The prior $P(h)$ “encodes the knowledge/preference” (s.18).

MAP vs MLE (s.19)

	MLE	MAP
Maximizes	likelihood $P(D	h)$
Uses prior?	No	Yes, $P(h)$
Formula	$h_{ML}=\arg\max P(D	h)$

The slide asks “When will MLE and MAP give the same results?”

Added context

When the prior $P(h)$ is uniform.

Classification use (s.20): $h_{MAP}=\arg {\max }_{h_i}p(h_i)p(d_1,\dots ,d_n|h_i)$, which needs too much data for large $n$ (Concept - Curse of Dimensionality); Concept - Naive Bayes fixes this.

Related

Concept - Bayes Rule, Concept - Maximum Likelihood Estimation, Concept - Prior Probability

Source: 03 Text Classification - Chunk 01 (s.1-30)