Bayesian Statistics: An Introduction, based on the author's third-year course on Bayesian statistics taught at the University of York, now reaches its fourth edition. It provides a concise account of the way in which the Bayesian approach to statistics develops and the contrast between it and the conventional approach. The theory is built up step by step, and important notions such as sufficiency are brought out of a discussion of the salient features of specific examples. This fourth edition includes significant new material on recent techniques such as variational methods, importance sampling, approximate computation and reversible jump MCMC. All chapters are updated, and the chapter on the Gibbs sampler is heavily expanded.