Electronic Journal of Probability

Discrepancy estimates for variance bounding Markov chain quasi-Monte Carlo

Josef Dick and Daniel Rudolf

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Markov chain Monte Carlo (MCMC) simulations are modeled as driven by true random numbers. We consider variance bounding Markov chains driven by a deterministic sequence of numbers. The star-discrepancy provides a measure of efficiency of such Markov chain quasi-Monte Carlo methods. We define a pull-back discrepancy of the driver sequence and state a close relation to the star-discrepancy of the Markov chain-quasi Monte Carlo samples. We prove that there exists a deterministic driver sequence such that the discrepancies decrease almost with the Monte Carlo rate $n^{-1/2}$. As for MCMC simulations,  a burn-in period can also be taken into account for Markov chain quasi-Monte Carlo to reduce the influence of the initial state. In particular, our discrepancy bound leads to an estimate of the error for the computation of expectations. To illustrate our theory we provide an example for the Metropolis algorithm based on a ball walk. Furthermore, under additional assumptions we prove the existence of a driver sequence such that the discrepancy of the corresponding deterministic Markov chain sample decreases with order $n^{-1+\delta}$ for every $\delta>0$.

Article information

Electron. J. Probab. Volume 19 (2014), paper no. 105, 24 pp.

Accepted: 5 November 2014
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J22: Computational methods in Markov chains [See also 65C40]
Secondary: 65C40: Computational Markov chains 62F15: Bayesian inference 65C05: Monte Carlo methods 60J05: Discrete-time Markov processes on general state spaces

Markov chain Monte Carlo Markov chain quasi-Monte Carlo variance bounding discrepancy theory spectral gap probabilistic method

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Dick, Josef; Rudolf, Daniel. Discrepancy estimates for variance bounding Markov chain quasi-Monte Carlo. Electron. J. Probab. 19 (2014), paper no. 105, 24 pp. doi:10.1214/EJP.v19-3132. https://projecteuclid.org/euclid.ejp/1465065747.

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