<Bayes theorem> gives
$$
\log p(\theta\mid y)=\log L(\theta)+\log\pi(\theta)-\log Z.
$$
Taking its posterior expectation after subtracting $\log\pi(\theta)$ yields
$$
D_{\mathrm{KL}}(p(\theta\mid y)\Vert\pi)
=\mathbb E_{\theta\mid y}\log L(\theta)-\log Z.
$$
Thus the equality holds for every proper prior and valid likelihood for which the displayed expectations are well-defined.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-219/4/a/ii/solution