Generalised Gibbs sampler and multigrid Monte Carlo for Bayesian computation

Although Monte Carlo methods have frequently been applied with success, indiscriminate use of Markov chain Monte Carlo leads to unsatisfactory performances in numerous applications. We present a generalised version of the Gibbs sampler that is based on conditional moves along the traces of groups of transformations in the sample space. We explore its connection with the multigrid Monte Carlo method and its use in designing more efficient samplers. The generalised Gibbs sampler provides a framework encompassing a class of recently proposed tricks such as parameter expansion and reparameterisation. To illustrate, we apply this new method to Bayesian inference problems for nonlinear state-space models, ordinal data and stochastic differential equations with discrete observations.