\[posterior \ odds=\frac{\pi(H_0|x)}{\pi(H_1|x)}=\frac{\pi(H_0)}{\pi(H_1)}\frac{f(x|H_0)}{f(x|H_1)}=\frac{\pi(H_0)}{\pi(H_1)} B = prior \ odds \ x \ Bayes \ factor\]

For composite hypothesis tests

For simple hypothesis test

\[B= \Lambda\]

\[\frac{\pi(H_0|x)}{\pi(H_1|x)}= \frac{\pi(H_0)}{\pi(H_1)} \Lambda\]

\[ log\frac{\pi(H_0|x)}{\pi(H_1|x)} = logit \left( \pi(H_0) \right)+ log \Lambda\]