= Solution
Use the orientation that favours the <null hypothesis>:
$$
\boxed{B_{01}(y)=\frac{p(y\mid H_0)}{p(y\mid H_1)}.}
$$
For a composite model the denominator is its <Bayesian model evidence>, obtained by integrating the <likelihood> against the model's <prior distribution>. This <Bayes factor> exceeds one when the observed data have greater marginal probability under $H_0$. By <Bayes' theorem>,
$$
\frac{\mathbb P(H_0\mid y)}{\mathbb P(H_1\mid y)}
=\frac{p(y\mid H_0)\pi_0/p(y)}{p(y\mid H_1)\pi_1/p(y)}
=\boxed{B_{01}(y)O_{01}}.
$$
Thus the <posterior odds> equal the <prior odds> multiplied by the <Bayes factor>. The reciprocal orientation instead gives $B_{10}=1/B_{01}$.
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