The Annals of Statistics

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

S. Chen, J. Dick, and A. B. Owen

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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.

Article information

Source
Ann. Statist. Volume 39, Number 2 (2011), 673-701.

Dates
First available in Project Euclid: 9 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aos/1299680951

Digital Object Identifier
doi:10.1214/10-AOS831

Mathematical Reviews number (MathSciNet)
MR2816335

Zentralblatt MATH identifier
1225.65010

Subjects
Primary: 65C40: Computational Markov chains 62F15: Bayesian inference
Secondary: 26A42: Integrals of Riemann, Stieltjes and Lebesgue type [See also 28-XX] 65C05: Monte Carlo methods

Keywords
Completely uniformly distributed coupling iterated function mappings Markov chain Monte Carlo

Citation

Chen, S.; Dick, J.; Owen, A. B. Consistency of Markov chain quasi-Monte Carlo on continuous state spaces. Ann. Statist. 39 (2011), no. 2, 673--701. doi:10.1214/10-AOS831. https://projecteuclid.org/euclid.aos/1299680951


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