Augmentation schemes for particle MCMC

Particle MCMC involves using a particle filter within an MCMC algorithm. For inference of a model which involves an unobserved stochastic process, the standard implementation uses the particle filter to propose new values for the stochastic process, and MCMC moves to propose new values for the parameters. We show how particle MCMC can be generalised beyond this. Our key idea is to introduce new latent variables. We then use the MCMC moves to update the latent variables, and the particle filter to propose new values for the parameters and stochastic process given the latent variables. A generic way of defining these latent variables is to model them as pseudo-observations of the parameters or of the stochastic process. By choosing the amount of information these latent variables have about the parameters and the stochastic process we can often improve the mixing of the particle MCMC algorithm by trading off the Monte Carlo error of the particle filter and the mixing of the MCMC moves. We show that using pseudo-observations within particle MCMC can improve its efficiency in certain scenarios: dealing with initialisation problems of the particle filter; speeding up the mixing of particle Gibbs when there is strong dependence between the parameters and the stochastic process; and enabling further MCMC steps to be used within the particle filter.

[1]  D. Blackwell,et al.  Ferguson Distributions Via Polya Urn Schemes , 1973 .

[2]  T. Ferguson.  A Bayesian Analysis of Some Nonparametric Problems , 1973 .

[3]  W. Wong,et al.  The calculation of posterior distributions by data augmentation , 1987 .

[4]  On Rates of Convergence of Stochastic Relaxation for Gaussian and Non-Gaussian Distributions* , 1991 .

[5]  Y. Amit.  On rates of convergence of stochastic relaxation for Gaussian and non-Gaussian distributions , 1991 .

[6]  Jun S. Liu,et al.  Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes , 1994 .

[7]  M. Pitt,et al.  Analytic Convergence Rates and Parameterization Issues for the Gibbs Sampler Applied to State Space Models , 1999 .

[8]  Jun S. Liu,et al.  Sequential Monte Carlo methods for dynamic systems , 1997 .

[9]  M. Pitt,et al.  Filtering via Simulation: Auxiliary Particle Filters , 1999 .

[10]  P. Fearnhead,et al.  An improved particle filter for non-linear problems , 1999 .

[11]  Simon J. Godsill,et al.  On sequential Monte Carlo sampling methods for Bayesian filtering , 2000, Stat. Comput..

[12]  P. Donnelly,et al.  Inference of population structure using multilocus genotype data. , 2000, Genetics.

[13]  P. Donnelly,et al.  Association mapping in structured populations. , 2000, American journal of human genetics.

[14]  J. Rosenthal,et al.  Optimal scaling for various Metropolis-Hastings algorithms , 2001 .

[15]  Xiao-Li Meng,et al.  The Art of Data Augmentation , 2001 .

[16]  Amir Dembo,et al.  Remarks on the maximum correlation coefficient , 2001 .

[17]  W. Gilks,et al.  Following a moving target—Monte Carlo inference for dynamic Bayesian models , 2001 .

[18]  Peter Donnelly,et al.  Assessing population differentiation and isolation from single‐nucleotide polymorphism data , 2002 .

[19]  Geir Storvik,et al.  Particle filters for state-space models with the presence of unknown static parameters , 2002, IEEE Trans. Signal Process..

[20]  M. Feldman,et al.  Genetic Structure of Human Populations , 2002, Science.

[21]  P. Fearnhead.  MCMC, sufficient statistics and particle filters. , 2002 .

[22]  P. Fearnhead.  Markov chain Monte Carlo, Sufficient Statistics, and Particle Filters , 2002 .

[23]  M. Stephens,et al.  Inference of population structure using multilocus genotype data: linked loci and correlated allele frequencies. , 2003, Genetics.

[24]  M. Stephens,et al.  Traces of Human Migrations in Helicobacter pylori Populations , 2003, Science.

[25]  Gareth O. Roberts,et al.  Non-centred parameterisations for hierarchical models and data augmentation. , 2003 .

[26]  P. Fearnhead,et al.  On‐line inference for hidden Markov models via particle filters , 2003 .

[27]  Paul Fearnhead,et al.  Particle filters for mixture models with an unknown number of components , 2004, Stat. Comput..

[28]  P. Moral.  Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications , 2004 .

[29]  D. Reich,et al.  Population Structure and Eigenanalysis , 2006, PLoS genetics.

[30]  D. Reich,et al.  Principal components analysis corrects for stratification in genome-wide association studies , 2006, Nature Genetics.

[31]  Christophe Andrieu,et al.  A tutorial on adaptive MCMC , 2008, Stat. Comput..

[32]  Darren J. Wilkinson,et al.  Discussion of Particle Markov chain Monte Carlo , 2008 .

[33]  Paul Fearnhead,et al.  Computational methods for complex stochastic systems: a review of some alternatives to MCMC , 2008, Stat. Comput..

[34]  Roman Holenstein,et al.  Particle Markov chain Monte Carlo , 2009 .

[35]  Robert B. Gramacy,et al.  Particle Learning of Gaussian Process Models for Sequential Design and Optimization , 2009, 0909.5262.

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

[37]  Nicholas G. Polson,et al.  Particle Learning and Smoothing , 2010, 1011.1098.

[38]  Nicholas G. Polson,et al.  Particle learning for general mixtures , 2010 .

[39]  David A. Rasmussen,et al.  Inference for Nonlinear Epidemiological Models Using Genealogies and Time Series , 2011, PLoS Comput. Biol..

[40]  Paul Fearnhead,et al.  MCMC for State–Space Models , 2011 .

[41]  Darren J Wilkinson,et al.  Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo , 2011, Interface Focus.

[42]  Tobias Rydén,et al.  Rao-Blackwellization of Particle Markov Chain Monte Carlo Methods Using Forward Filtering Backward Sampling , 2011, IEEE Transactions on Signal Processing.

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

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

[45]  Christophe Andrieu,et al.  Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers , 2013, 1312.6432.

[46]  John Parslow,et al.  On Disturbance State-Space Models and the Particle Marginal Metropolis-Hastings Sampler , 2012, SIAM/ASA J. Uncertain. Quantification.

[47]  Sumeetpal S. Singh,et al.  On the Particle Gibbs Sampler , 2013 .

[48]  Robert Kohn,et al.  On general sampling schemes for Particle Markov chain Monte Carlo methods , 2014 .

[49]  Frank D. Wood,et al.  A New Approach to Probabilistic Programming Inference , 2014, AISTATS.

[50]  P. Moral,et al.  On Feynman-Kac and particle Markov chain Monte Carlo models , 2014 .

[51]  P. Fearnhead,et al.  Particle Metropolis-adjusted Langevin algorithms , 2014, 1412.7299.

[52]  Fredrik Lindsten,et al.  Particle gibbs with ancestor sampling , 2014, J. Mach. Learn. Res..

[53]  P. Fearnhead,et al.  Particle Metropolis adjusted Langevin algorithms for state-space models , 2014, 1402.0694.

[54]  Christian P. Robert,et al.  An introduction to the special issue “Joint IMS-ISBA meeting - MCMSki 4” , 2015, Stat. Comput..

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

[56]  Christophe Andrieu,et al.  Introduction to “Particle Metropolis-Hastings using gradient and Hessian information” by J. Dahlin, F. Lindsten, T. Schön , 2015, Stat. Comput..

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