Chapter 8

Advanced Topics in MCMC

The theory and practice of Markov chain Monte Carlo continues to advance at a rapid pace. Two particularly notable innovations are the dimension shifting reversible jump MCMC method and approaches for adapting proposal distributions while the algorithm is running. Also, applications for Bayesian inference continue to be of broad interest. In this chapter we survey a variety of higher level MCMC methods and explore some of the possible uses of MCMC to solve challenging statistical problems.

Sections 8.1–8.5 introduce a wide variety of advanced MCMC topics, including adaptive, reversible jump, and auxiliary variable MCMC, additional Metropolis–Hasting methods, and perfect sampling methods. In Section 8.6 we discuss an application of MCMC to maximum likelihood estimation. We conclude the chapter in Section 8.7 with an example where several of these methods are applied to facilitate Bayesian inference for spatial or image data.

8.1 Adaptive MCMC

One challenge with MCMC algorithms is that they often require tuning to improve convergence behavior. For example, in a Metropolis–Hastings algorithm with a normally distributed proposal distribution, some trial and error is usually required to tune the variance of the proposal distribution to achieve an optimal acceptance rate (see Section 7.3.1.3). Tuning the proposal distribution becomes even more challenging when the number of parameters is large. Adaptive MCMC (AMCMC) algorithms allow for automatic tuning ...

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