Difficulties in the use of auxiliary variables in Markov chain Monte Carlo methods

Markov chain Monte Carlo (MCMC) methods are now widely used in a diverse range of application areas to tackle previously intractable problems. Difficult questions remain, however, in designing MCMC samplers for problems exhibiting severe multimodality where standard methods may exhibit prohibitively slow movement around the state space. Auxiliary variable methods, sometimes together with multigrid ideas, have been proposed as one possible way forward. Initial disappointing experiments have led to data-driven modifications of the methods. In this paper, these suggestions are investigated for lattice data such as is found in imaging and some spatial applications. The results suggest that adapting the auxiliary variables to the specific application is beneficial. However the form of adaptation needed and the extent of the resulting benefits are not always clear-cut.