Competing two hypotheses

The idea of competing two hypotheses …. [UNDER CONSTRUCTION]

A simple formula for the posterior odds:

$$\underset{\text{posterior for }{H}_a}{\underbrace{odds(H_a|\text{obs})}}=\underset{\text{Likelihood ratio}}{\underbrace{\left[\frac{{\mathcal{L}}_a(\text{obs})}{{\mathcal{L}}_0(\text{obs})}\right]}}\times \underset{\text{prior for }{H}_a}{\underbrace{odds(H_a)}}$$