The idea of competing two hypotheses …. [UNDER CONSTRUCTION]
A simple formula for the posterior odds:
\[\underbrace{odds(H_a | \text{obs})}_{\Large\text{posterior for } H_a} = \underbrace{\left[\frac{{\cal L}_a(\text{obs})}{{\cal L}_0(\text{obs})} \right]}_{\Large\text{Likelihood ratio}}\ \times\ \underbrace{odds(H_a)}_{\Large\text{prior for } H_a}\]