Analysis of the Ensemble and Polynomial Chaos Kalman Filters in Bayesian Inverse Problems

TL;DR

This paper investigates the theoretical properties of Ensemble and Polynomial Chaos Kalman filters in Bayesian inverse problems, revealing their convergence behavior and relation to linear estimators, which impacts their use in uncertainty quantification.

Contribution

It provides a new interpretation of these filters in the Bayesian context and proves their convergence to a specific analysis random variable, highlighting limitations for uncertainty quantification.

Findings

Both filters converge to an analysis random variable as ensemble size or polynomial degree increases.

The analysis variable is more akin to a linear Bayes estimator than the true Bayesian posterior.

Limited applicability of these filters for accurate uncertainty quantification in inverse problems.

Abstract

We analyze the Ensemble and Polynomial Chaos Kalman filters applied to nonlinear stationary Bayesian inverse problems. In a sequential data assimilation setting such stationary problems arise in each step of either filter. We give a new interpretation of the approximations produced by these two popular filters in the Bayesian context and prove that, in the limit of large ensemble or high polynomial degree, both methods yield approximations which converge to a well-defined random variable termed the analysis random variable. We then show that this analysis variable is more closely related to a specific linear Bayes estimator than to the solution of the associated Bayesian inverse problem given by the posterior measure. This suggests limited or at least guarded use of these generalized Kalman filter methods for the purpose of uncertainty quantification.