Ensemble-Conditional Gaussian Processes (Ens-CGP): Representation, Geometry, and Inference
Sai Ravela, Jae Deok Kim, Kenneth Gee, Xingjian Yan, Samson Mercier, Lubna Albarghouty, Anamitra Saha
TL;DR
The paper unifies Gaussian conditioning, RKHS regularization, and Kalman-type inference within a single discrete object: the conditional Gaussian process (CGP). By introducing Ens-CGP, it treats ensemble moments as a finite-dimensional Gaussian prior and shows that the Kalman analysis update, MAP/QP, and RKHS regression are all manifestations of the same conditioning rule under linear-Gaussian models. The key contributions include a rigorous discrete equivalence chart, a representation–computation separation, and the demonstration that ensembles act as low-rank priors rather than standalone Kalman objects. This framework clarifies when iterative ensemble methods are valid probabilistic updates and when they function merely as optimization surrogates, with broad relevance to inverse problems, data assimilation, and kernel-based regression. It provides a transferable backbone connecting classical and modern approaches across geophysics, climate science, and ML.
Abstract
We formulate Ensemble-Conditional Gaussian Processes (Ens-CGP), a finite-dimensional synthesis that centers ensemble-based inference on the conditional Gaussian law. Conditional Gaussian processes (CGP) arise directly from Gaussian processes under conditioning and, in linear-Gaussian settings, define the full posterior distribution for a Gaussian prior and linear observations. Classical Kalman filtering is a recursive algorithm that computes this same conditional law under dynamical assumptions; the conditional Gaussian law itself is therefore the underlying representational object, while the filter is one computational realization. In this sense, CGP provides the probabilistic foundation for Kalman-type methods as well as equivalent formulations as a strictly convex quadratic program (MAP estimation), RKHS-regularized regression, and classical regularization. Ens-CGP is the ensemble instantiation of this object, obtained by treating empirical ensemble moments as a (possibly low-rank) Gaussian prior and performing exact conditioning. By separating representation (GP -> CGP -> Ens-CGP) from computation (Kalman filters, EnKF variants, and iterative ensemble schemes), the framework links an earlier-established representational foundation for inference to ensemble-derived priors and clarifies the relationships among probabilistic, variational, and ensemble perspectives.