Bayes Learning - MAP hypothesis

Question

Suppose I have a set of hypotheses $H = \{h_1, h_2\}$ mutual exclusive. For them $P(h_1) = 0.2$ and $P(h_2) = 0.3$ (prior distribution). Suppose we know also that

$$P(Y=0 | h_1) = 0.2$$ $$P(Y=0 | h_2) = 0.4$$

where $Y$ is an attribute (target) that can have two values $\{1,0\}$. Suppose finally that you observe the event $Y = 0$.

Which one is the MAP (Maximum a posteriori) hypothesis?

• MAP is $h_1$
• MAP is $h_2$
• there's no enough element to find MAP
• MAP $h_1$ = MAP $h_2$
• nobody of the possible answer above

Answer 1

Your question is basic application of the Bayes Theorem

$$P(h_1|Y=0) = \frac{P(Y=0|h_1)P(h_1)}{P(Y=0)}=\frac{0.04}{P(Y=0)}$$ $$P(h_2|Y=0) = \frac{P(Y=0|h_2)P(h_2)}{P(Y=0)}=\frac{0.12}{P(Y=0)}$$

So the $h_2$ is the more probable hypothesis, as $P(Y=0)>0$