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

Discard burn-in from the MCMC output and retain the sampled $\Delta t$ coordinate. A normalized histogram or kernel density estimation of these draws approximates $p(\Delta t\mid D)$. Autocorrelation changes the Monte Carlo uncertainty, so uncertainty bands should use the chain's effective sample size rather than its raw length.