Exact affine conditioning beyond Gaussians: a unique characterization of the ensemble Kalman update

Abstract

The analysis step of the stochastic ensemble Kalman filter, called the ensemble Kalman update (EnKU), is widely used for approximating posterior distributions in inverse problems and data assimilation. The EnKU approximates the posterior distribution $\pi_{X\mid Y=y_\star}$ by pushing forward the joint distribution $(X,Y)\sim\pi$ through an affine map $L^{\mathrm{EnKU}}_{\pi,y_\star}(x,y)$ that depends only on the covariance structure of $\pi$ and the observation $y_\star$. While the EnKU yields the exact posterior for Gaussian $\pi$ in the mean-field, this property alone does not uniquely determine the EnKU. In fact, there are infinitely many affine maps $L_{\pi, y_\star}$ that achieve such exact conditioning. In this paper, we offer a novel characterization of the EnKU among all such affine maps. We first exhaustively characterize the set ${E}^{\mathrm{EnKU}}$ of joint distributions for which the EnKU yields exact conditioning, showing that it is much larger than the set of Gaussians. Next, we show that except for a small class of highly symmetric distributions within ${E}^{\mathrm{EnKU}}$, the EnKU is the {unique} exact affine conditioning map. Further, we characterize the largest possible set of distributions ${F}$ for which a distribution-dependent, weakly observation-dependent, affine map exists, a class of transports that naturally includes the EnKU. We show that ${F}={E}^{\mathrm{EnKU}}\cup{S}_{\mathrm{nl-dec}}$ with a small symmetry class ${S}_{\mathrm{nl-dec}}$, meaning that for affine conditioning beyond the Gaussian setting, the EnKU has an exact set that is essentially maximally large.