Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 219 / 3 / c / Solution

Let $\pi$ denote the posterior and let the proposal density satisfy $q({\theta }^{\prime }\mid \theta )=q(\theta \mid {\theta }^{\prime })$. For distinct states, the Metropolis transition density isConsequentlyThe rejection mass on the diagonal also satisfies detailed balance, so the posterior is invariant.