Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 219 / 4 / a / ii

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- Solution ii

Solution

Bayes theorem gives

lo g p (θ ∣ y) = lo g L (θ) + lo g π (θ) - lo g Z .

Taking its posterior expectation after subtracting

lo g π (θ)

yields

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

Thus the equality holds for every proper prior and valid likelihood for which the displayed expectations are well-defined.

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