Quantitative Wasserstein Propagation of Chaos for Transport Ensemble Filters

2026-06-23

AI Summary

This work addresses the lack of non-asymptotic, high-probability convergence guarantees for transport-based ensemble filters (TEFs) in approximating the state distribution of hidden Markov models. By constructing a probabilistic analysis framework, the study characterizes the chaotic propagation behavior of TEFs and identifies their mean-field limiting dynamics. Leveraging synchronous coupling, conditional moment and tail stability analyses, and empirical Wasserstein distance estimates, the paper establishes, for the first time, pathwise Wasserstein convergence for TEFs, proving that the particle system converges with high probability at the Monte Carlo rate to i.i.d. samples from the mean-field limit. The theory unifies algorithms such as the Ensemble Kalman Filter (EnKF) and the Ensemble Square Root Filter (EnSRF), yielding explicit non-asymptotic convergence bounds.

Abstract

We develop a general probabilistic framework for analyzing propagation of chaos in transport ensemble filters (TEFs), a broad class of interacting particle systems that are used to approximate the sequence of state distributions in hidden Markov models given a history of observations. This class of transport-based filtering algorithms includes the widely used ensemble Kalman filter (EnKF), based on affine updates at each filtering step, as well as the ensemble stochastic map filter (EnSMF), which employs nonlinear updates. For this class, we identify the limiting mean-field dynamics. We then establish non-asymptotic, high-probability, pathwise Wasserstein convergence of the interacting particle system to an i.i.d. ensemble drawn from this mean-field limit at the Monte Carlo rate. Convergence to the mean-field law itself follows with the usual dimension-dependent empirical Wasserstein rate. The proof combines a synchronous coupling construction with stability of moments and tails under conditioning, together with quantitative estimates for the propagation of the underlying dynamics through the interacting particle system. Applying our theory to both the EnKF and the EnSMF yields the first non-asymptotic, high-probability convergence guarantees for TEFs.

Problem

• propagation of chaos
• transport ensemble filters
• Wasserstein convergence
• interacting particle systems
• hidden Markov models

Innovation

• Wasserstein convergence
• propagation of chaos
• transport ensemble filters
• mean-field limit
• non-asymptotic guarantees