### 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 language | English (US) |
---|---|

Article number | 7040041 |

Pages (from-to) | 4185-4190 |

Number of pages | 6 |

Journal | Proceedings of the IEEE Conference on Decision and Control |

Volume | 2015-February |

Issue number | February |

DOIs | |

State | Published - Jan 1 2014 |

Event | 2014 53rd IEEE Annual Conference on Decision and Control, CDC 2014 - Los Angeles, United States Duration: Dec 15 2014 → Dec 17 2014 |

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### ASJC Scopus subject areas

- Control and Systems Engineering
- Modeling and Simulation
- Control and Optimization

### Cite this

*Proceedings of the IEEE Conference on Decision and Control*,

*2015-February*(February), 4185-4190. [7040041]. https://doi.org/10.1109/CDC.2014.7040041

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

Research output: Contribution to journal › Conference article

*Proceedings of the IEEE Conference on Decision and Control*, vol. 2015-February, no. February, 7040041, pp. 4185-4190. https://doi.org/10.1109/CDC.2014.7040041

}

TY - JOUR

T1 - Poisson's equation in nonlinear filtering

AU - Laugesen, Richard

AU - Mehta, Prashant G.

AU - Meyn, Sean P.

AU - Raginsky, Maxim

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84988286468&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84988286468&partnerID=8YFLogxK

U2 - 10.1109/CDC.2014.7040041

DO - 10.1109/CDC.2014.7040041

M3 - Conference article

AN - SCOPUS:84988286468

VL - 2015-February

SP - 4185

EP - 4190

JO - Proceedings of the IEEE Conference on Decision and Control

JF - Proceedings of the IEEE Conference on Decision and Control

SN - 0191-2216

IS - February

M1 - 7040041

ER -