Nonasymptotic bounds on the estimation error of MCMC algorithms

Abstract

We address the problem of upper bounding the mean square error of MCMC estimators. Our analysis is nonasymptotic. We first establish a general result valid for essentially all ergodic Markov chains encountered in Bayesian computation and a possibly unbounded target function $f$. The bound is sharp in the sense that the leading term is exactly ${\sigma }_{\mathrm{as}}^2(P,f)/n$, where ${\sigma }_{\mathrm{as}}^2(P,f)$ is the CLT asymptotic variance. Next, we proceed to specific additional assumptions and give explicit computable bounds for geometrically and polynomially ergodic Markov chains under quantitative drift conditions. As a corollary, we provide results on confidence estimation.

Published in at http://dx.doi.org/10.3150/12-BEJ442 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm). arXiv admin note: text overlap with arXiv:0907.4915

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• Quantitative bounds on convergence of time-inhomogeneous Markov chains
• Concentration inequalities for dependent Random variables via the martingale method
• Curvature, concentration and error estimates for Markov chain Monte Carlo
• Practical drift conditions for subgeometric rates of convergence
• Optimal Hoeffding bounds for discrete reversible Markov chains
• Explicit error bounds for lazy reversible Markov Chain Monte Carlo
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• Error bounds of MCMC for functions with unbounded stationary variance
• Computation of expectations by Markov chain Monte Carlo methods
• Multilevel Monte Carlo estimation of the expected value of sample information
• On a Metropolis-Hastings importance sampling estimator
• Dimension-independent Markov chain Monte Carlo on the sphere
• Almost sure convergence rates of adaptive increasingly rare Markov chain Monte Carlo
• Optimal convergence rates of MCMC integration for functions with unbounded second moment