# Poisson's equation in nonlinear filtering

arXiv: https://arxiv.org/abs/1412.5845 · HTML: https://mlsanity.com/paper/1412.5845/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Dec 2014.

## Built on by (most cited first, 10 of 10)

- [Sampling in Unit Time with Kernel Fisher-Rao Flow](https://mlsanity.com/paper/2401.03892/) (2024)
- [Gradient Flows for Sampling: Mean-Field Models, Gaussian Approximations and Affine Invariance](https://mlsanity.com/paper/2302.11024/) (2023)
- [An Optimal Transport Formulation of Bayes' Law for Nonlinear Filtering Algorithms](https://mlsanity.com/paper/2203.11869/) (2022)
- [Stein transport for Bayesian inference](https://mlsanity.com/paper/2409.01464/) (2024)
- [Connections between sequential Bayesian inference and evolutionary dynamics](https://mlsanity.com/paper/2411.16366/) (2024)
- [Learning Energy-Based Generative Models via Potential Flow: A Variational Principle Approach to Probability Density Homotopy Matching](https://mlsanity.com/paper/2504.16262/) (2025)
- [Variational Gaussian approximation of the Kushner optimal filter](https://mlsanity.com/paper/2310.01859/) (2023)
- [A Deep Probabilistic Flow-Based Framework for Unsupervised Cross-Domain Soft Sensing](https://mlsanity.com/paper/2306.04919/) (2023)
- [Variational Potential Flow: A Novel Probabilistic Framework for Energy-Based Generative Modelling](https://mlsanity.com/paper/2407.15238/) (2024)
- [Functional role of synchronization: A mean-field control perspective](https://mlsanity.com/paper/2502.00590/) (2025)

## Abstract (arXiv)

The aim of this paper is to provide a variational interpretation of the nonlinear filter in continuous time. A time-stepping procedure is introduced, consisting of successive minimization problems in the space of probability densities. The weak form of the nonlinear filter is derived via analysis of the first-order optimality conditions for these problems. The derivation shows the nonlinear filter dynamics may be regarded as a gradient flow, or a steepest descent, for a certain energy functional with respect to the Kullback-Leibler divergence. The second part of the paper is concerned with derivation of the feedback particle filter algorithm, based again on the analysis of the first variation. The algorithm is shown to be exact. That is, the posterior distribution of the particle matches exactly the true posterior, provided the filter is initialized with the true prior.