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

With $D=(y_1,y_2,t)$ and $\nu =(\Delta m,c,A,\tau )$,A random-walk Metropolis algorithm proposes ${\theta }^{\prime }=\theta +Z$ from a symmetric multivariate normal increment, conveniently using logarithmic coordinates for $A,\tau >0$, and accepts with probabilityAfter burn-in, retain a suitably long chain and assess convergence and effective sample size of a Markov chain.