Ensemble-Conditional Gaussian Processes (Ens-CGP): Representation, Geometry, and Inference

TL;DR

This paper introduces Ensemble-Conditional Gaussian Processes (Ens-CGP), a novel ensemble-based inference framework grounded in the conditional Gaussian law, linking classical methods with modern ensemble techniques.

Contribution

Ens-CGP provides a unified, finite-dimensional ensemble framework for conditional Gaussian processes, clarifying relationships among probabilistic, variational, and ensemble inference methods.

Findings

Ens-CGP generalizes Kalman filtering and ensemble methods.

It separates representation from computation in Gaussian process inference.

Framework links probabilistic and ensemble perspectives clearly.

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.