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Quantitative non-geometric convergence bounds for independence samplers

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Roberts, Gareth O. and Rosenthal, Jeffrey S. (Jeffrey Seth) (2009) Quantitative non-geometric convergence bounds for independence samplers. Working Paper. University of Warwick. Centre for Research in Statistical Methodology, Coventry.

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Abstract

Markov chain Monte Carlo (MCMC) algorithms are widely used in statistics, physics, and computer science, to sample from complicated high-dimensional probability distributions. A central question is how quickly the chain converges to the target (stationarity) distribution. In this paper, we consider this question for a particular class of MCMC algorithms, independence samplers (Hastings, 1970; Tierney, 1994).

Item Type: Working or Discussion Paper (Working Paper)
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH): Distribution (Probability theory), Markov processes, Monte Carlo method
Series Name: Working papers
Publisher: University of Warwick. Centre for Research in Statistical Methodology
Place of Publication: Coventry
Date: 2009
Volume: Vol.2009
Number: No.9
Number of Pages: 14
Status: Not Peer Reviewed
Access rights to Published version: Open Access
References: P.H. Baxendale (2005), Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Prob. 15, 700–738. N.H. Bingham, C.M. Goldie, and J.L. Teugels (1987), Regular Variation. Cambridge University Press, Cambridge. R. Douc, E. Moulines, and J.S. Rosenthal (2004), Quantitative bounds on convergence of time-inhomogeneous Markov chains. Ann. Appl. Prob. 14, 1643–1665. R. Douc, E. Moulines, and P. Soulier (2007), Computable convergence rates for sub-geometric ergodic Markov chains. Bernoulli 13, 831–848. G. Fort and E. Moulines (2000), Computable Bounds For Subgeometrical And Geometrical Ergodicity. Available at: http://citeseer.ist.psu.edu/fort00computable.html G. Fort and E. Moulines (2003), Polynomial ergodicity of Markov transition kernels. Stoch. Proc. Appl. 103, 57–99. C. Geyer (1992), Practical Markov chain Monte Carlo. Stat. Sci., Vol. 7, No. 4, 473–483. W.K. Hastings (1970), Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109. S.F. Jarner and G.O. Roberts (2002), Polynomial convergence rates of Markov chains. Ann. Appl. Prob., 224–247, 2002. G.L. Jones and J.P. Hobert (2001), Honest exploration of intractable probability distributions via Markov chain Monte Carlo. Stat. Sci. 16, 312–334. G.L. Jones and J.P. Hobert (2004). Sufficient burn-in for Gibbs samplers for a hierarchical random effects model. Ann. Stat. 32, 784–817. J. Liu (1996), Metropolized independent sampling with comparisons to rejection sampling and importance sampling. Stat. and Comput. 6, 113–119. D. Marchev and J.P. Hobert (2004), Geometric ergodicity of van Dyk and Meng’s algorithm for the multivariate Student’s t model. J. Amer. Stat. Assoc. 99, 228–238. K. L. Mengersen and R. L. Tweedie (1996), Rates of convergence of the Hastings and Metropolis algorithms. Ann. Stat. 24, 101–121. S.P. Meyn and R.L. Tweedie (1993), Markov chains and stochastic stability. Springer-Verlag, London. Available at: http://probability.ca/MT/ G.O. Roberts (1999), A note on acceptance rate criteria for CLTs for Metropolis-Hastings algorithms. J. Appl. Prob. 36, 1210–1217. G.O. Roberts and J.S. Rosenthal (1998), Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion). Can. J. Stat. 26, 5–31. G.O. Roberts and J.S. Rosenthal (2004), General state space Markov chains and MCMC algorithms. Prob. Surv. 1, 20–71. G.O. Roberts and R.L. Tweedie (1996), Geometric Convergence and Central Limit Theorems for Multidimensional Hastings and Metropolis Algorithms. Biometrika 83, 95–110. J.S. Rosenthal (1995), Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Stat. Assoc. 90, 558–566. J.S. Rosenthal (1997), Independence sampler Java applet. Available at: http://probability.ca/jeff/java/exp.html J.S. Rosenthal (2002), Quantitative convergence rates of Markov chains: A simple account. Electronic Comm. Prob. 7, 123–128. R.L. Smith and L. Tierney (1996), Exact transition probabilities for the independence Metropolis sampler. Preprint. L. Tierney (1994), Markov chains for exploring posterior distributions (with discussion). Ann. Stat. 22, 1701–1762. S. Wolfram (1988), Mathematica: A system for doing mathematics by computer. Addison- Wesley, New York.
URI: http://wrap.warwick.ac.uk/id/eprint/35200

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