Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 47 / 5 / a / i / Solution

A systematic Gibbs sampler updates the coordinates of $\theta$ successively from their full conditional distributions. At iteration $t$, drawFor a single coordinate, if the current state has distribution $\pi$, its untouched coordinates have the correct marginal law and the resampled coordinate has precisely the conditional law under $\pi$. Integrating marginal times conditional therefore recovers $\pi$. Each coordinate kernel preserves this invariant distribution, so their composition also preserves it. This proves the target of the Gibbs sampler; the whole systematic sweep need not be a reversible kernel.

Starting from a suitable state and assuming the required irreducibility and convergence conditions, discard an initial transient and retain subsequent states as a dependent sample from, or approximately from, $\pi$. In finite-state problems irreducibility and aperiodicity suffice; in continuous-state problems the corresponding positive recurrence and ergodicity conditions must hold. Successive draws are generally dependent, so Monte Carlo uncertainty must account for that dependence.