Quasi Markov Chain Monte Carlo Methods
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Quasi-Monte Carlo (QMC) methods for estimating integrals are attractive since the resulting estimators converge at a faster rate than pseudo-random Monte Carlo. However, they can be difficult to set up on arbitrary posterior densities within the Bayesian framework, in particular for inverse problems. We introduce a general parallel Markov chain Monte Carlo (MCMC) framework, for which we prove a law of large numbers and a central limit theorem. We further extend this approach to the use of adaptive kernels and state conditions, under which ergodicity holds. As a further extension, an importance sampling estimator is derived, for which asymptotic unbiasedness is proven. We consider the use of completely uniformly distributed (CUD) numbers and non-reversible transitions within the above stated methods, which leads to a general parallel quasi-MCMC (QMCMC) methodology. We prove consistency of the resulting estimators and demonstrate numerically that this approach scales close to n^-1 as we increase parallelisation, instead of the usual n^-1/2 that is typical of standard MCMC algorithms. In practical statistical models we observe up to 2 orders of magnitude improvement compared with pseudo-random methods.
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