# An Optimal Transport Formulation of Bayes' Law for Nonlinear Filtering Algorithms

arXiv: https://arxiv.org/abs/2203.11869 · HTML: https://mlsanity.com/paper/2203.11869/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 41 references that are in the map.
- First posted on arXiv 22 Mar 2022.

## Builds on (most cited first, 15 of 15)

- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Optimal Control Via Neural Networks: A Convex Approach](https://mlsanity.com/paper/1805.11835/) (2018)
- [Conditional Simulation Using Diffusion Schr"odinger Bridges](https://mlsanity.com/paper/2202.13460/) (2022)
- [Coupling techniques for nonlinear ensemble filtering](https://mlsanity.com/paper/1907.00389/) (2019)
- [Preconditioned training of normalizing flows for variational inference in inverse problems](https://mlsanity.com/paper/2101.03709/) (2021)
- [An introduction to sampling via measure transport](https://mlsanity.com/paper/1602.05023/) (2016)
- [The efficacy and generalizability of conditional GANs for posterior inference in physics-based inverse problems](https://mlsanity.com/paper/2202.07773/) (2022)
- [Gradient Flows in Filtering and Fisher-Rao Geometry](https://mlsanity.com/paper/1710.00064/) (2017)
- [Poisson's equation in nonlinear filtering](https://mlsanity.com/paper/1412.5845/) (2014)
- [Gradient Flows in Uncertainty Propagation and Filtering of Linear Gaussian Systems](https://mlsanity.com/paper/1704.00102/) (2017)
- [Optimal Transportation Methods in Nonlinear Filtering: The feedback particle filter](https://mlsanity.com/paper/2102.10712/) (2021)
- [Proximal Recursion for the Wonham Filter](https://mlsanity.com/paper/1909.05827/) (2019)
- [An Optimal Transport Formulation of the Ensemble Kalman Filter](https://mlsanity.com/paper/1910.02338/) (2019)

## Built on by (most cited first, 11 of 11)

- [Sampling in Unit Time with Kernel Fisher-Rao Flow](https://mlsanity.com/paper/2401.03892/) (2024)
- [Conditional Optimal Transport on Function Spaces](https://mlsanity.com/paper/2311.05672/) (2023)
- [Reliable amortized variational inference with physics-based latent distribution correction](https://mlsanity.com/paper/2207.11640/) (2022)
- [Nonlinear Filtering with Brenier Optimal Transport Maps](https://mlsanity.com/paper/2310.13886/) (2023)
- [A generative flow for conditional sampling via optimal transport](https://mlsanity.com/paper/2307.04102/) (2023)
- [Exact nonlinear state estimation](https://mlsanity.com/paper/2310.10976/) (2023)
- [Fast filtering of non-Gaussian models using Amortized Optimal Transport Maps](https://mlsanity.com/paper/2503.12633/) (2025)
- [Closed-form conditional diffusion models for data assimilation](https://mlsanity.com/paper/2603.21291/) (2026)
- [Energy-based Transport for Amortized Bayesian Inference](https://mlsanity.com/paper/2605.15407/) (2026)
- [A lift for input-convex neural net training](https://mlsanity.com/paper/2605.24274/) (2026)
- [Generative models for simulation based filtering: Formulations and Empirical Comparisons](https://mlsanity.com/paper/2609.16317/) (2026)

## Abstract (arXiv)

This paper presents a variational representation of the Bayes' law using optimal transportation theory. The variational representation is in terms of the optimal transportation between the joint distribution of the (state, observation) and their independent coupling. By imposing certain structure on the transport map, the solution to the variational problem is used to construct a Brenier-type map that transports the prior distribution to the posterior distribution for any value of the observation signal. The new formulation is used to derive the optimal transport form of the Ensemble Kalman filter (EnKF) for the discrete-time filtering problem and propose a novel extension of EnKF to the non-Gaussian setting utilizing input convex neural networks. Finally, the proposed methodology is used to derive the optimal transport form of the feedback particle filler (FPF) in the continuous-time limit, which constitutes its first variational construction without explicitly using the nonlinear filtering equation or Bayes' law.