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Paper No. 11-23

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K Latuszynski, B Miasojewdow and W Niemiro

Nonasymptotic bounds on the estimation error of MCMC algorithms

Asbstract: We address the problem of upper bounding the mean square error of MCMC estimators. Our analysis is non-asymptotic.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 _2 as(P; f)=n, where _2 as(P; f) is the CLT asymptotic variance. Next, we proceed to specific assumptions and give explicit computable bounds for geometrically and polynomially ergodic Markov chains. As a corollary we provide results on confidence estimation. AMS 2000 subject classifications: Primary 60J05, 65C05; secondary 62F15.

Keywords: Mean square error, Computable bounds, Geometric ergodicity, Polynomial ergodicity, Drift conditions, Regeneration, Asymptotic variance, Confidence estimation.