Markov chain Monte Carlo, Sufficient Statistics, and Particle Filters

This article considers how to implement Markov chain Monte Carlo (MCMC) moves within a particle filter. Previous, similar, attempts have required the complete history (“trajectory”) of each particle to be stored. Here, it is shown how certain MCMC moves can be introduced within a particle filter when only summaries of each particles' trajectory are stored. These summaries are based on sufficient statistics. Using this idea, the storage requirement of the particle filter can be substantially reduced, and MCMC moves can be implemented more efficiently. We illustrate how this idea can be used for both the bearingsonly tracking problem and a model of stochastic volatility. We give a detailed comparison of the performance of different particle filters for the bearings-only tracking problem. MCMC, combined with a sensible initialization of the filter and stratified resampling, produces substantial gains in the efficiency of the particle filter.

[1]  S. Taylor.  Financial Returns Modelled by the Product of Two Stochastic Processes , 1961 .

[2]  V. Aidala.  Kalman Filter Behavior in Bearings-Only Tracking Applications , 1979, IEEE Transactions on Aerospace and Electronic Systems.

[3]  V. Aidala,et al.  Utilization of modified polar coordinates for bearings-only tracking , 1983 .

[4]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[5]  Jun S. Liu,et al.  Blind Deconvolution via Sequential Imputations , 1995 .

[6]  N. Peach,et al.  Bearings-only tracking using a set of range-parameterised extended Kalman filters , 1995 .

[7]  G. Kitagawa.  Monte Carlo Filter and Smoother for Non-Gaussian Nonlinear State Space Models , 1996 .

[8]  Peter Green,et al.  Markov chain Monte Carlo in Practice , 1996 .

[9]  N. Shephard.  Statistical aspects of ARCH and stochastic volatility , 1996 .

[10]  R.A. Iltis,et al.  A consistent estimation criterion for multisensor bearings-only tracking , 1996, IEEE Transactions on Aerospace and Electronic Systems.

[11]  Sampling strategies for Monte Carlo filters of non-linear systems , 1996 .

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

[13]  M. Pitt,et al.  Likelihood analysis of non-Gaussian measurement time series , 1997 .

[14]  N. G. Best,et al.  Dynamic conditional independence models and Markov chain Monte Carlo methods , 1997 .

[15]  J. Cadre,et al.  Bearings-only tracking for maneuvering sources , 1998 .

[16]  M. Pitt,et al.  Time Varying Covariances: A Factor Stochastic Volatility Approach (with discussion , 1998 .

[17]  G. Peters,et al.  Monte Carlo Approximations for General State-Space Models , 1998 .

[18]  P. Fearnhead,et al.  Improved particle filter for nonlinear problems , 1999 .

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

[20]  P. Moral,et al.  Central limit theorem for nonlinear filtering and interacting particle systems , 1999 .

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

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

[23]  Walter R. Gilks,et al.  RESAMPLE-MOVE Filtering with Cross-Model Jumps , 2001, Sequential Monte Carlo Methods in Practice.

[24]  R. Wishner,et al.  Utilization of Modified Polar Coordinates for Bearings-Only Tracking , 2001 .

[25]  N. Chopin.  A sequential particle filter method for static models , 2002 .

[26]  Timothy J. Robinson,et al.  Sequential Monte Carlo Methods in Practice , 2003 .