Dimension-free Wasserstein contraction of nonlinear filters

TL;DR

This paper establishes dimension-free Wasserstein contraction properties for nonlinear filters in partially observed diffusions, under specific conditions on the signal dynamics and likelihood functions, applicable to various models including linear-Gaussian and neural spike-train.

Contribution

It provides the first dimension-independent contraction results for nonlinear filters with affine drift and log-concave likelihoods, encompassing both ergodic and nonergodic cases.

Findings

Filter stability holds without observation assumptions in several models.

Contraction rate is independent of the state-space dimension.

Applicable to models like stochastic volatility and neural spike-train.

Abstract

For a class of partially observed diffusions, conditions are given for the map from the initial condition of the signal to filtering distribution to be contractive with respect to Wasserstein distances, with rate which does not necessarily depend on the dimension of the state-space. The main assumptions are that the signal has affine drift and constant diffusion coefficient and that the likelihood functions are log-concave. Ergodic and nonergodic signals are handled in a single framework. Examples include linear-Gaussian, stochastic volatility, neural spike-train and dynamic generalized linear models. For these examples filter stability can be established without any assumptions on the observations.