Accelerating pseudo-marginal Metropolis-Hastings by correlating auxiliary variables

Pseudo-marginal Metropolis-Hastings (pmMH) is a powerful method for Bayesian inference in models where the posterior distribution is analytical intractable or computationally costly to evaluate directly. It operates by introducing additional auxiliary variables into the model and form an extended target distribution, which then can be evaluated point-wise. In many cases, the standard Metropolis-Hastings is then applied to sample from the extended target and the sought posterior can be obtained by marginalisation. However, in some implementations this approach suers from poor mixing as the auxiliary variables are sampled from an independent proposal. We propose a modication to the pmMH algorithm in which a Crank-Nicolson (CN) proposal is used instead. This results in that we introduce a positive correlation in the auxiliary variables. We investigate how to tune the CN proposal and its impact on the mixing of the resulting pmMH sampler. The conclusion is that the proposed modication

[1]  Csaba Kelemen,et al.  Simple and Robust Mutation Strategy for Metropolis Light Transport Algorithm , 2001 .

[2]  A. Doucet,et al.  A Tutorial on Particle Filtering and Smoothing: Fifteen years later , 2008 .

[3]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[4]  N. Chopin,et al.  Sequential Quasi-Monte Carlo , 2014, 1402.4039.

[5]  Thomas B. Schön,et al.  Robust auxiliary particle filters using multiple importance sampling , 2014, 2014 IEEE Workshop on Statistical Signal Processing (SSP).

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

[7]  Johan Dahlin,et al.  Quasi-Newton particle Metropolis-Hastings , 2015, 1502.03656.

[8]  G. Roberts,et al.  MCMC Methods for Functions: ModifyingOld Algorithms to Make Them Faster , 2012, 1202.0709.

[9]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[10]  Antonietta Mira,et al.  Zero variance Markov chain Monte Carlo for Bayesian estimators , 2010, Stat. Comput..

[11]  A. Doucet,et al.  The correlated pseudomarginal method , 2015, Journal of the Royal Statistical Society: Series B (Statistical Methodology).

[12]  N. Shephard,et al.  BAYESIAN INFERENCE BASED ONLY ON SIMULATED LIKELIHOOD: PARTICLE FILTER ANALYSIS OF DYNAMIC ECONOMIC MODELS , 2011, Econometric Theory.

[13]  John G. Kemeny,et al.  Finite Markov chains , 1960 .

[14]  P. Moral,et al.  Sequential Monte Carlo samplers , 2002, cond-mat/0212648.

[15]  Thomas Bo Schön,et al.  Efficient approximate Bayesian inference for models with intractable likelihoods , 2015 .

[16]  A. Stuart,et al.  Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions , 2011, 1112.1392.

[17]  G. Roberts,et al.  MCMC methods for diffusion bridges , 2008 .

[18]  L. Tierney.  Markov Chains for Exploring Posterior Distributions , 1994 .

[19]  Carlos E. Rodríguez,et al.  Searching for efficient Markov chain Monte Carlo proposal kernels , 2013, Proceedings of the National Academy of Sciences.

[20]  M. Pitt,et al.  Particle filters for continuous likelihood evaluation and maximisation , 2011 .

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

[22]  Christian P. Robert,et al.  Monte Carlo Statistical Methods , 2005, Springer Texts in Statistics.

[23]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

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

[25]  Anthony Lee,et al.  Towards smooth particle filters for likelihood estimation with multivariate latent variables , 2008 .

[26]  Leonidas J. Guibas,et al.  Metropolis light transport , 1997, SIGGRAPH.

[27]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[28]  M. Pitt,et al.  Importance Sampling Squared for Bayesian Inference in Latent Variable Models , 2013, 1309.3339.

[29]  Ralph S. Silva,et al.  On Some Properties of Markov Chain Monte Carlo Simulation Methods Based on the Particle Filter , 2012 .

[30]  Jean-Michel Marin,et al.  Approximate Bayesian computational methods , 2011, Statistics and Computing.

[31]  Ajay Jasra,et al.  Approximate Bayesian Computation for a Class of Time Series Models , 2014, 1401.0265.

[32]  Johan Dahlin,et al.  Particle Metropolis–Hastings using gradient and Hessian information , 2013, Statistics and Computing.

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

[34]  M. Villani,et al.  Bayesian optimisation for fast approximate inference in state-space models with intractable likelihoods , 2015, 1506.06975.

[35]  Loukia Meligkotsidou,et al.  Augmentation schemes for particle MCMC , 2014, Statistics and Computing.

[36]  Toshiya Hachisuka,et al.  Multiplexed metropolis light transport , 2014, ACM Trans. Graph..

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