Analysis of the Ensemble and Polynomial Chaos Kalman Filters in Bayesian Inverse Problems
For practitioners using Kalman filter variants in Bayesian inverse problems, this work reveals a fundamental limitation of these methods for uncertainty quantification.
The paper analyzes Ensemble and Polynomial Chaos Kalman filters for nonlinear Bayesian inverse problems, proving that in the large ensemble or high polynomial degree limit, both methods converge to a random variable that is more related to a linear Bayes estimator than to the true posterior, suggesting caution in using these filters for uncertainty quantification.
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.