 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/iii/paper-36/1/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Use the orientation that favours the null hypothesis:

B_{01} (y) = \frac{p ( y ∣ H _{0} )}{p ( y ∣ 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}

\frac{P ( H _{0} ∣ y )}{P ( H _{1} ∣ y )} = \frac{p ( y ∣ H _{0} ) π _{0} / p ( y )}{p ( y ∣ H _{1} ) π _{1} / p ( y )} = 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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