Poisson's equation in nonlinear filtering

Research output: Chapter in Book/Report/Conference proceedingConference contribution


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)
Title of host publication53rd IEEE Conference on Decision and Control,CDC 2014
PublisherInstitute of Electrical and Electronics Engineers Inc.
Number of pages6
ISBN (Electronic)9781479977468
StatePublished - 2014
Event2014 53rd IEEE Annual Conference on Decision and Control, CDC 2014 - Los Angeles, United States
Duration: Dec 15 2014Dec 17 2014

Publication series

NameProceedings of the IEEE Conference on Decision and Control
ISSN (Print)0743-1546
ISSN (Electronic)2576-2370


Other2014 53rd IEEE Annual Conference on Decision and Control, CDC 2014
Country/TerritoryUnited States
CityLos Angeles

ASJC Scopus subject areas

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


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