Establishing some order amongst exact approximations of MCMCs

Exact approximations of Markov chain Monte Carlo (MCMC) algorithms are a general emerging class of sampling algorithms. One of the main ideas behind exact approximations consists of replacing intractable quantities required to run standard MCMC algorithms, such as the target probability density in a Metropolis-Hastings algorithm, with estimators. Perhaps surprisingly, such approximations lead to powerful algorithms which are exact in the sense that they are guaranteed to have correct limiting distributions. In this paper we discover a general framework which allows one to compare, or order, performance measures of two implementations of such algorithms. In particular, we establish an order with respect to the mean acceptance probability, the first autocorrelation coefficient, the asymptotic variance and the right spectral gap. The key notion to guarantee the ordering is that of the convex order between estimators used to implement the algorithms. We believe that our convex order condition is close to optimal, and this is supported by a counter-example which shows that a weaker variance order is not sufficient. The convex order plays a central role by allowing us to construct a martingale coupling which enables the comparison of performance measures of Markov chain with differing invariant distributions, contrary to existing results. We detail applications of our result by identifying extremal distributions within given classes of approximations, by showing that averaging replicas improves performance in a monotonic fashion and that stratification is guaranteed to improve performance for the standard implementation of the Approximate Bayesian Computation (ABC) MCMC method.

[1]  Paul Fearnhead,et al.  Constructing Summary Statistics for Approximate Bayesian Computation: Semi-automatic ABC , 2010, 1004.1112.

[2]  A. Doucet,et al.  Efficient implementation of Markov chain Monte Carlo when using an unbiased likelihood estimator , 2012, 1210.1871.

[3]  L. Tierney A note on Metropolis-Hastings kernels for general state spaces , 1998 .

[4]  Werner Hürlimann Extremal Moment Methods and Stochastic Orders.Application in Actuarial Science: Chapters IV, V and VI , 2008 .

[5]  J. Rosenthal,et al.  On the efficiency of pseudo-marginal random walk Metropolis algorithms , 2013, The Annals of Statistics.

[6]  J. Rosenthal,et al.  Geometric Ergodicity and Hybrid Markov Chains , 1997 .

[7]  Marc Goovaerts,et al.  Dependency of Risks and Stop-Loss Order , 1996, ASTIN Bulletin.

[8]  P. Peskun,et al.  Optimum Monte-Carlo sampling using Markov chains , 1973 .

[9]  D. Rudolf,et al.  Explicit error bounds for Markov chain Monte Carlo , 2011, 1108.3201.

[10]  D. Balding,et al.  Approximate Bayesian computation in population genetics. , 2002, Genetics.

[11]  C. Fox,et al.  Coupled MCMC with a randomized acceptance probability , 2012, 1205.6857.

[12]  Christophe Andrieu,et al.  Annealed Importance Sampling Reversible Jump MCMC Algorithms , 2013 .

[13]  Werner Hürlimann,et al.  Extremal Moment Methods and Stochastic Orders , 2008 .

[14]  G. Roberts,et al.  Adaptive Markov Chain Monte Carlo through Regeneration , 1998 .

[15]  A. Sapozhnikov Subgeometric rates of convergence of f-ergodic Markov chains , 2006 .

[16]  A. Müller,et al.  Comparison Methods for Stochastic Models and Risks , 2002 .

[17]  C. Andrieu,et al.  Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms , 2012, 1210.1484.

[18]  Antonietta Mira,et al.  AN EXTENSION OF PESKUN AND TIERNEY ORDERINGS TO CONTINUOUS TIME MARKOV CHAINS , 2008 .

[19]  Galin L. Jones On the Markov chain central limit theorem , 2004, math/0409112.

[20]  P. Donnelly,et al.  Inferring coalescence times from DNA sequence data. , 1997, Genetics.

[21]  Samuel Karlin,et al.  Generalized convex inequalities , 1963 .

[22]  S. F. Jarner,et al.  Geometric ergodicity of Metropolis algorithms , 2000 .

[23]  J. Shanthikumar,et al.  Multivariate Stochastic Orders , 2007 .

[24]  Gerrit K. Janssens,et al.  On the use of bounds on the stop-loss premium for an inventory management decision problem , 2008 .

[25]  Paul Marjoram,et al.  Markov chain Monte Carlo without likelihoods , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[26]  R. Douc,et al.  Practical drift conditions for subgeometric rates of convergence , 2004, math/0407122.

[27]  C. Robert,et al.  Controlled MCMC for Optimal Sampling , 2001 .

[28]  P. Fearnhead,et al.  Exact and computationally efficient likelihood‐based estimation for discretely observed diffusion processes (with discussion) , 2006 .

[29]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[30]  R. Douc,et al.  Comparison of asymptotic variances of inhomogeneous Markov chains with application to Markov chain Monte Carlo methods , 2013, 1307.3719.

[31]  Richard Bellman,et al.  Some inequalities for the square root of a positive definite matrix , 1968 .

[32]  V. Strassen The Existence of Probability Measures with Given Marginals , 1965 .

[33]  W. Hoeffding On the Distribution of the Number of Successes in Independent Trials , 1956 .

[34]  Christopher C. Drovandi,et al.  Pseudo-marginal algorithms with multiple CPUs , 2014 .

[35]  C. Andrieu,et al.  The pseudo-marginal approach for efficient Monte Carlo computations , 2009, 0903.5480.

[36]  M. Beaumont Estimation of population growth or decline in genetically monitored populations. , 2003, Genetics.

[37]  Heikki Haario,et al.  Efficient MCMC for Climate Model Parameter Estimation: Parallel Adaptive Chains and Early Rejection , 2012 .

[38]  K. Latuszy'nski,et al.  Nonasymptotic bounds on the estimation error of MCMC algorithms , 2011, 1106.4739.

[39]  R. Tweedie,et al.  Geometric L 2 and L 1 convergence are equivalent for reversible Markov chains , 2001, Journal of Applied Probability.

[40]  S. Caracciolo,et al.  Nonlocal Monte Carlo algorithm for self-avoiding walks with fixed endpoints , 1990 .

[41]  Marco Scarsini,et al.  Stochastic Comparisons of Symmetric Sampling Designs , 2012 .

[42]  Marco Scarsini,et al.  Stochastic comparisons of stratified sampling techniques for some Monte Carlo estimators , 2011 .

[43]  G. Roberts,et al.  Polynomial convergence rates of Markov chains. , 2002 .

[44]  Marc Goovaerts,et al.  Analytical best upper bounds on stop-loss premiums , 1982 .

[45]  D. Ceperley,et al.  The penalty method for random walks with uncertain energies , 1998, physics/9812035.

[46]  A. Doucet,et al.  Particle Markov chain Monte Carlo methods , 2010 .