Quasi-Monte Carlo (QMC) methods for estimating integrals are attractive since the resulting estimators typically 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 propose a principled and efficient way of applying QMC to drive algorithms based on a general parallel Markov chain Monte Carlo (MCMC) framework, in which multiple proposals are generated per iteration.
We provide numerous methodological extensions of the original algorithm, including the use of non-reversible
transition kernels, adaptive proposal kernels and Rao-Blackwellisation. Further, we prove a law of large numbers
and a central limit theorem, ergodicity of the proposed adaptive methods and asymptotic unbiasedness for
estimates based on the Rao-Blackwellisation scheme.
We consider the use of completely uniformly distributed (CUD) numbers within the previously stated algorithms,
which leads to a general parallel Markov chain quasi-Monte Carlo (MCQMC) methodology. A scheme that
efficiently produces CUD seeds for arbitrary problem dimensions and proposal numbers based on an already
existing CUD sequence is developed. We prove ergodicity of the resulting adaptive methods and asymptotic
unbiasedness for the Rao-Blackwellised estimates. For the latter, we demonstrate numerically in a number of
statistical models that this approach scales close to n^{−2} as we increase parallelisation, instead of the usual
n^{−1} that is typical of pseudo-random MCMC algorithms. The improved rate is proven theoretically in a special case. Simulations are performed for Bayesian linear and logistic regression, simple non-linear ODE models and a
complex model for cardiac excitation. The CUD driven Rao-Blackwellisation algorithms yield multiple orders
of magnitude reduction in the variance and MSE of the resulting estimates compared to their pseudo-driven
counterparts and to reference algorithms such as Metropolis-Hastings.