Consistency of Markov chain quasi-Monte Carlo on continuous state spaces

TL;DR

This paper extends the theoretical understanding of using quasi-Monte Carlo sequences in Markov chain Monte Carlo methods for continuous distributions, demonstrating their consistency and potential for improved simulation accuracy.

Contribution

It provides the first theoretical justification for the consistency of quasi-Monte Carlo sequences in continuous-state MCMC algorithms, beyond discrete cases.

Findings

Quasi-Monte Carlo sequences yield consistent estimates in continuous MCMC.

Theoretical support for replacing pseudo-random numbers with low-discrepancy sequences.

Results also confirm the consistency of standard pseudo-random number methods.

Abstract

The random numbers driving Markov chain Monte Carlo (MCMC) simulation are usually modeled as independent U(0,1) random variables. Tribble [Markov chain Monte Carlo algorithms using completely uniformly distributed driving sequences (2007) Stanford Univ.] reports substantial improvements when those random numbers are replaced by carefully balanced inputs from completely uniformly distributed sequences. The previous theoretical justification for using anything other than i.i.d. U(0,1) points shows consistency for estimated means, but only applies for discrete stationary distributions. We extend those results to some MCMC algorithms for continuous stationary distributions. The main motivation is the search for quasi-Monte Carlo versions of MCMC. As a side benefit, the results also establish consistency for the usual method of using pseudo-random numbers in place of random ones.