  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.
