Short Course

This short course introduces rigorous MCMC for infinite-dimensional Bayesian models. We begin with a concise review of Bayesian inference and classical MCMC, including Gibbs sampling and Metropolis–Hastings algorithms, then move to probability on function spaces, focusing on Gaussian process priors and geostatistical models.

The core develops algorithms that are formally infinite-dimensional: targets are defined as measures on function spaces, and the samplers operate without discretization. This avoids bias from mesh approximations and can yield significant statistical and computational benefits.

We introduce retrospective sampling to formalize updates on infinite-dimensional state spaces and use it to present two main families: (i) models in which the infinite-dimensional component has a closed-form full conditional, enabling exact Gibbs steps; and (ii) Metropolis–Hastings with preconditioned Crank–Nicolson proposals that preserve Gaussian reference measures.

The final module applies these ideas to specific geostatistical models, highlighting the structure of the resulting algorithms and implementation details.