Paul Fearnhead is Professor of Statistics at Lancaster University, with research interests in Bayesian and Computational Statistics. He has been awarded Cambridge University's Adams prize, and the Guy Medals in Bronze and Silver from the Royal Statistical Society. He was elected a fellow of the International Society for Bayesian Analysis in 2024 and is currently the Editor of Biometrika. Christopher Nemeth is Professor of Statistics at Lancaster University, working at the interface of Statistics and Machine Learning, with a focus on probabilistic modelling and the development of new computational tools for statistical inference. In 2020, he was awarded a UKRI Turing AI Fellowship to develop new algorithms for probabilistic AI. Chris. J. Oates leads a team working in the areas of Computational Statistics and Probabilistic Machine Learning at Newcastle University. He was awarded a Leverhulme Prize for Mathematics and Statistics in 2023, and the Guy Medal in Bronze of the Royal Statistical Society in 2024. Chris Sherlock is Professor of Statistics at Lancaster University. After working in data assimilation, numerical modelling and software engineering, he was caught up in the excitement of Computationally Intensive Bayesian Statistics, obtaining a Ph.D. in the topic and now leading a group of like-minded researchers.

• Preface; 1. Background; 2. Reversible MCMC and its Scaling; 3. Stochastic Gradient MCMC Algorithms; 4. Non-Reversible MCMC; 5. Continuous-Time MCMC; 6. Assessing and Improving MCMC; References; Index.