Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 219 / 4 / f

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- Solution f

Solution

It holds generally whenever the posterior is well defined and the expectations exist. Bayes' theorem gives

lo g \frac{p ( θ ∣ y )}{π ( θ )} = lo g L (θ) - lo g Z .

Taking posterior expectations yields

D_{KL} {p (θ ∣ y) ∥ π} = E_{θ ∣ y} lo g L (θ) - lo g Z,

which rearranges to the claimed evidence decomposition.

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