  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
introduce a general parallel Markov chain Monte Carlo (MCMC) framework, for
which we prove a law of large numbers and a central limit theorem. In that
context, non-reversible transitions are investigated. We then 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 within the above mentioned
algorithms, 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 $n^{-2}$ as we increase parallelisation,
instead of the usual $n^{-1}$ that is typical of standard MCMC algorithms. In
practical statistical models we observe multiple orders of magnitude
improvement compared with pseudo-random methods.
