Poisson's equation in nonlinear filtering

Research output: Contribution to journalConference article

Abstract

The goal of this paper is to gain insight into the equations arising in nonlinear filtering, as well as into the feedback particle filter introduced in recent research. The analysis is inspired by the optimal transportation literature and by prior work on variational formulation of nonlinear filtering. The construction involves a discrete-time recursion based on the successive solution of minimization problems involving the so-called forward variational representation of the elementary Bayes' formula. The construction shows that the dynamics of the nonlinear filter may be regarded as a gradient flow, or a steepest descent, for a certain energy functional with respect to the Kullback-Leibler divergence pseudo-metric. The feedback particle filter algorithm is obtained using similar analysis. This filter is a controlled system, where the control is obtained via consideration of the first order optimality conditions for the variational problem. The filter is shown to be exact, i.e., the posterior distribution of the particle matches exactly the true posterior, provided the filter is initialized with the true prior.

Original languageEnglish (US)
Article number7040041
Pages (from-to)4185-4190
Number of pages6
JournalProceedings of the IEEE Conference on Decision and Control
Volume2015-February
Issue numberFebruary
DOIs
StatePublished - Jan 1 2014
Event2014 53rd IEEE Annual Conference on Decision and Control, CDC 2014 - Los Angeles, United States
Duration: Dec 15 2014Dec 17 2014

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Nonlinear filtering
Nonlinear Filtering
Poisson equation
Poisson's equation
Particle Filter
Filter
Feedback
Optimal Transportation
First-order Optimality Conditions
Bayes' Formula
Pseudometric
Kullback-Leibler Divergence
Nonlinear Filters
Gradient Flow
Steepest Descent
Variational Formulation
Energy Functional
Posterior distribution
Variational Problem
Recursion

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization

Cite this

Poisson's equation in nonlinear filtering. / Laugesen, Richard; Mehta, Prashant G.; Meyn, Sean P.; Raginsky, Maxim.

In: Proceedings of the IEEE Conference on Decision and Control, Vol. 2015-February, No. February, 7040041, 01.01.2014, p. 4185-4190.

Research output: Contribution to journalConference article

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