Bayes Theorem and MAP Hypotheses

$$P(h\vert D) = \frac{P(D\vert h)\cdot P(h)}{P(D)}$$ (1)

Thus

 

$$\equiv \underset{h \in H}{\rm argmax}\quad P(h\vert D)$$ h_MAP

$$\Rightarrow h_{MAP} \equiv \underset{h \in H}{\rm argmax}\quad \frac{P(D\vert h) \cdot P(h)}{P(D)}$$

$$\equiv \underset{h \in H}{\rm argmax}\quad P(D\vert h)\cdot P(h)$$ (2)

Sometimes, we have no grounds to suppose that any particular hypothesis in H is more likely than any other. Then we can assume a uniform prior. In this case Eqn 2 simplifies further to

 

$$\equiv \underset{h \in H}{\rm argmax}\quad P(D\vert h)$$ h_ML (3)

Here, h_ML is called the Maximum Likelihood hypothesis. A ML hypothesis is a MAP hypothesis that assumes a uniform prior over the space of possible hypotheses.