Variational Markov Chain Monte Carlo for Bayesian smoothing of non-linear niffusions

Y. Shen, D. Cornford, M. Opper, C. Archambeau

Research output: Working paperTechnical report

Abstract

In this paper we develop set of novel Markov chain Monte Carlo algorithms for Bayesian smoothing of partially observed non-linear diffusion processes. The sampling algorithms developed herein use a deterministic approximation to the posterior distribution over paths as the proposal distribution for a mixture of an independence and a random walk sampler. The approximating distribution is sampled by simulating an optimized time-dependent linear diffusion process derived from the recently developed variational Gaussian process approximation method. Flexible blocking strategies are introduced to further improve mixing, and thus the efficiency, of the sampling algorithms. The algorithms are tested on two diffusion processes: one with double-well potential drift and another with SINE drift. The new algorithm's accuracy and efficiency is compared with state-of-the-art hybrid Monte Carlo based path sampling. It is shown that in practical, finite sample, applications the algorithm is accurate except in the presence of large observation errors and low observation densities, which lead to a multi-modal structure in the posterior distribution over paths. More importantly, the variational approximation assisted sampling algorithm outperforms hybrid Monte Carlo in terms of computational efficiency, except when the diffusion process is densely observed with small errors in which case both algorithms are equally efficient.
Original languageEnglish
Place of PublicationBirmingham (UK)
PublisherAston University
Number of pages25
Publication statusUnpublished - 19 Jan 2010

Publication series

NameTechnical Report
PublisherNeural Computing Research Group
Volume8106

Fingerprint

Markov processes
Sampling
Computational efficiency

Keywords

  • Markov chain Monte Carlo algorithms
  • non-linear diffusion processes
  • variational Gaussian process approximation
  • hybrid Monte Carlo

Cite this

Shen, Y., Cornford, D., Opper, M., & Archambeau, C. (2010). Variational Markov Chain Monte Carlo for Bayesian smoothing of non-linear niffusions. (Technical Report; Vol. 8106). Birmingham (UK): Aston University.
Shen, Y. ; Cornford, D. ; Opper, M. ; Archambeau, C. / Variational Markov Chain Monte Carlo for Bayesian smoothing of non-linear niffusions. Birmingham (UK) : Aston University, 2010. (Technical Report).
@techreport{6322c46097da47f49be9f6b863e398fe,
title = "Variational Markov Chain Monte Carlo for Bayesian smoothing of non-linear niffusions",
abstract = "In this paper we develop set of novel Markov chain Monte Carlo algorithms for Bayesian smoothing of partially observed non-linear diffusion processes. The sampling algorithms developed herein use a deterministic approximation to the posterior distribution over paths as the proposal distribution for a mixture of an independence and a random walk sampler. The approximating distribution is sampled by simulating an optimized time-dependent linear diffusion process derived from the recently developed variational Gaussian process approximation method. Flexible blocking strategies are introduced to further improve mixing, and thus the efficiency, of the sampling algorithms. The algorithms are tested on two diffusion processes: one with double-well potential drift and another with SINE drift. The new algorithm's accuracy and efficiency is compared with state-of-the-art hybrid Monte Carlo based path sampling. It is shown that in practical, finite sample, applications the algorithm is accurate except in the presence of large observation errors and low observation densities, which lead to a multi-modal structure in the posterior distribution over paths. More importantly, the variational approximation assisted sampling algorithm outperforms hybrid Monte Carlo in terms of computational efficiency, except when the diffusion process is densely observed with small errors in which case both algorithms are equally efficient.",
keywords = "Markov chain Monte Carlo algorithms, non-linear diffusion processes, variational Gaussian process approximation, hybrid Monte Carlo",
author = "Y. Shen and D. Cornford and M. Opper and C. Archambeau",
year = "2010",
month = "1",
day = "19",
language = "English",
series = "Technical Report",
publisher = "Aston University",
type = "WorkingPaper",
institution = "Aston University",

}

Shen, Y, Cornford, D, Opper, M & Archambeau, C 2010 'Variational Markov Chain Monte Carlo for Bayesian smoothing of non-linear niffusions' Technical Report, vol. 8106, Aston University, Birmingham (UK).

Variational Markov Chain Monte Carlo for Bayesian smoothing of non-linear niffusions. / Shen, Y.; Cornford, D.; Opper, M.; Archambeau, C.

Birmingham (UK) : Aston University, 2010. (Technical Report; Vol. 8106).

Research output: Working paperTechnical report

TY - UNPB

T1 - Variational Markov Chain Monte Carlo for Bayesian smoothing of non-linear niffusions

AU - Shen, Y.

AU - Cornford, D.

AU - Opper, M.

AU - Archambeau, C.

PY - 2010/1/19

Y1 - 2010/1/19

N2 - In this paper we develop set of novel Markov chain Monte Carlo algorithms for Bayesian smoothing of partially observed non-linear diffusion processes. The sampling algorithms developed herein use a deterministic approximation to the posterior distribution over paths as the proposal distribution for a mixture of an independence and a random walk sampler. The approximating distribution is sampled by simulating an optimized time-dependent linear diffusion process derived from the recently developed variational Gaussian process approximation method. Flexible blocking strategies are introduced to further improve mixing, and thus the efficiency, of the sampling algorithms. The algorithms are tested on two diffusion processes: one with double-well potential drift and another with SINE drift. The new algorithm's accuracy and efficiency is compared with state-of-the-art hybrid Monte Carlo based path sampling. It is shown that in practical, finite sample, applications the algorithm is accurate except in the presence of large observation errors and low observation densities, which lead to a multi-modal structure in the posterior distribution over paths. More importantly, the variational approximation assisted sampling algorithm outperforms hybrid Monte Carlo in terms of computational efficiency, except when the diffusion process is densely observed with small errors in which case both algorithms are equally efficient.

AB - In this paper we develop set of novel Markov chain Monte Carlo algorithms for Bayesian smoothing of partially observed non-linear diffusion processes. The sampling algorithms developed herein use a deterministic approximation to the posterior distribution over paths as the proposal distribution for a mixture of an independence and a random walk sampler. The approximating distribution is sampled by simulating an optimized time-dependent linear diffusion process derived from the recently developed variational Gaussian process approximation method. Flexible blocking strategies are introduced to further improve mixing, and thus the efficiency, of the sampling algorithms. The algorithms are tested on two diffusion processes: one with double-well potential drift and another with SINE drift. The new algorithm's accuracy and efficiency is compared with state-of-the-art hybrid Monte Carlo based path sampling. It is shown that in practical, finite sample, applications the algorithm is accurate except in the presence of large observation errors and low observation densities, which lead to a multi-modal structure in the posterior distribution over paths. More importantly, the variational approximation assisted sampling algorithm outperforms hybrid Monte Carlo in terms of computational efficiency, except when the diffusion process is densely observed with small errors in which case both algorithms are equally efficient.

KW - Markov chain Monte Carlo algorithms

KW - non-linear diffusion processes

KW - variational Gaussian process approximation

KW - hybrid Monte Carlo

M3 - Technical report

T3 - Technical Report

BT - Variational Markov Chain Monte Carlo for Bayesian smoothing of non-linear niffusions

PB - Aston University

CY - Birmingham (UK)

ER -

Shen Y, Cornford D, Opper M, Archambeau C. Variational Markov Chain Monte Carlo for Bayesian smoothing of non-linear niffusions. Birmingham (UK): Aston University. 2010 Jan 19. (Technical Report).