= Posterior odds
{title2=$O_{01}(y)=O_{01}B_{01}(y)$}
For two exhaustive mutually exclusive models, posterior odds are the ratio of their <posterior probabilities>. <Bayes' theorem> gives
$$
\frac{\mathbb P(H_0\mid y)}{\mathbb P(H_1\mid y)}
=\frac{\mathbb P(H_0)}{\mathbb P(H_1)}\frac{p(y\mid H_0)}{p(y\mid H_1)}.
$$
The second factor is the <Bayes factor>, formed from model <prior predictive distributions>. Evidence favouring one model need not make that model's posterior probability exceed one half if its prior probability was sufficiently small.
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