Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 36 / 4 / a

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Solution

Compute the Bayesian model evidence for each model by integrating its sampling density against its parameter prior:

m_{i} (y) = \int p_{Y} (y ∣ ψ_{i}, M_{i}) p (ψ_{i} ∣ M_{i}) d ψ_{i} .

The posterior odds are the prior odds multiplied by the Bayes factor:

\frac{Pr ( M _{1} ∣ y )}{Pr ( M _{2} ∣ y )} = \frac{Pr ( M _{1} )}{Pr ( M _{2} )} \frac{m _{1} ( y )}{m _{2} ( y )} .

The parameter priors used for evidence must be normalized; arbitrary normalization constants in model-specific improper priors would make this comparison undefined.

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