Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 219 / 2 / c / Solution

The posterior isA random-walk Metropolis–Hastings algorithm proposes ${\vartheta }^{\prime }$ from a symmetric density about the current $\vartheta =(\mu ,\theta )$ and accepts with probabilityFor distinct states, multiplying the transition density by the target density giveswhich is symmetric and proves detailed balance. Run multiple dispersed chains, tune proposals during warm-up, inspect traces, effective sample sizes and convergence diagnostics, then estimate the period mean by averaging $2\pi /{\omega }_m$ over retained draws.