Choosing a Hypothesis

Generally want the most probable hypothesis given the training data.
The maximum a posteriori hypothesis is $h_{MAP}$ where \[ \array{ h_{MAP} & = \arg \max_{h \in H} P(h\,|\,D) \\ & = \arg \max_{h \in H} \frac{P(D\,|\,h) P(h)}{P(D)} \\ & = \arg \max_{h \in H}P(D\,|\,h) P(h) } \] we dropped $P(D)$ because its independent of $h$.
If we assume that all hypothesis have the same prior probability, that is $\forall_{i \neq j}P(h_i) = P(h_j)$, then we can simplify even more and choose the maximum likelihood hypothesis \[h_{ML} = \arg \max_{h \in H} P(D\,|\,h) \]