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.

翻译：我们提出了集成条件高斯过程（Ens-CGP），这是一种有限维的合成方法，将基于集成的推断建立在条件高斯法则之上。条件高斯过程（CGP）直接来源于高斯过程在条件作用下的结果，在线性高斯设定中，它定义了高斯先验与线性观测下的完整后验分布。经典卡尔曼滤波是一种在动态假设下计算同一条件法则的递归算法；因此，条件高斯法则本身是底层的表示对象，而滤波器是其一种计算实现。从这个意义上说，CGP为卡尔曼型方法以及等价表述（如严格凸二次规划（MAP估计）、RKHS正则化回归和经典正则化）提供了概率论基础。Ens-CGP是该对象的集成实例化，通过将经验集成矩视为（可能低秩的）高斯先验并执行精确条件化而得到。通过将表示（GP -> CGP -> Ens-CGP）与计算（卡尔曼滤波器、EnKF变体及迭代集成方案）分离，该框架将先前建立的推断表示基础与集成导出的先验联系起来，并阐明了概率、变分和集成视角之间的关系。