Steinwart proves with a Banach-space-valued martingale method that conditional distributions of jointly Gaussian variables stay Gaussian and are approximated by finite-dimensional filtering sequences
Synopsis
The work studies conditional distributions of two Banach-space-valued jointly Gaussian random variables, showing they remain Gaussian and can be determined by a finite-dimensional approximation scheme based on filtering sequences: conditional means converge in the E-norm, covariance operators converge in nuclear norm, conditional probabilities converge weakly, and for continuous Gaussian processes conditioned on partial infinite path observations the mean and covariance functions converge uniformly.
Interpretation
For Banach-space-valued X and finite-dimensional Y, the conditional distribution PX|Y(·|y) is almost surely Gaussian with mean and covariance given via cov(X,Y)(cov Y)†, and Z=µX|Y∘Y is a version of E(X|Y) satisfying cov(Z)=cov(X)−CX|Y and G(Z)⊂G(Y). Previously [8, Corollary 3.10.3] only guaranteed a Gaussian version without mean/covariance formulas, and [56, 58] required a representing sequence for cov(Y); this work gives explicit formulas for general separable Banach E and finite-dimensional Y and proves the distributional properties of Z. Theorem 3.1 and its proof (Section 6.1) verify the mean and covariance formulas via a w∗-dense dual sequence, the finite-dimensional Theorem 2.3, and the transformation theorem D.2 for regular conditional probabilities, using Hahn-Banach and uniqueness of abstract covariance operators (Lemma B.2).
For infinite-dimensional Y, a filtering sequence (An) with Yn=An∘Y is introduced, and PX|Yn(·|An(y)) is shown to converge weakly to PX|Y(·|y), with means converging in the E-norm and covariance operators in nuclear norm, while PX|Y(·|y) is Gaussian. Previously the Gaussianity of infinite-dimensional conditional distributions was known, but mean/covariance formulas for non-Hilbert F relied on hard-to-obtain representing sequences; this work reduces the infinite-dimensional problem to computable finite-dimensional projection limits and gives nuclear-norm-level covariance convergence. Theorem 3.3 combines Theorem 3.1, the martingale convergence Theorem D.5, the integral convergence Theorem 6.3 for regular conditional probabilities, and uniform tightness to prove items i)–viii), with covariance convergence in the nuclear norm ∥·∥nuc.
Every separable Banach space admits a filtering sequence, and concrete constructions are given for Hilbert spaces, RKHSs, C(T), C1([0,1]), and general Banach spaces consisting of functions; for separable BSFs a pseudo-metric can be built so that point evaluations separate. The filtering-sequence notion was not previously systematized; this work translates the measure-theoretic condition into w∗-dense or separating dual sequences (Propositions 4.1, 4.3) and proves general existence (Theorem 4.2) and function-space constructions (Theorems 4.5, 4.9, Lemma 4.10). Propositions 4.1, 4.3, Theorems 4.2, 4.5, 4.9 and Corollaries 4.4, 4.6, 4.7 are all proved, relying on metrizability of the w∗-topology (Theorem A.1) and continuity of point evaluations.
Applied to continuous Gaussian processes: for partial infinite observations Y=X|S on closed S⊂T, the conditional mean and covariance functions are given by uniformly convergent limits of finite-observation formulas, with mX|Y=g(s)=g(s) and kX|Y(s,t)=0 at observation points, and conditional probabilities converge weakly. Previously [35] required S=T and additional assumptions on the observational model; this work allows general closed S⊂T without extra observational-model assumptions and provides the weak convergence (38). Theorems 5.2 and 5.3 follow from Theorems 3.1, 3.3 and Lemma 8.1, with convergence uniform in t over C(T) and in t1,t2 for the covariance.
Perspective
The results apply to jointly Gaussian variables on separable Banach spaces E and F, and to finite-dimensional observations induced by filtering sequences; for continuous Gaussian processes, they apply to compact metric T and closed S⊂T with partial observations. They enable researchers to construct computable finite-dimensional approximations on (reproducing kernel) Hilbert spaces, C(T), C1([0,1]), and general function Banach spaces using point evaluations or derivative functionals, and thereby obtain convergence of conditional means, covariances, and conditional probabilities. For GP4ML practitioners, this means the textbook conditioning formulas on C(T) indeed correspond to regular conditional probabilities and can be approximated by finite observation sequences.
The work provides convergence and existence results but no convergence rates; the choice of filtering sequence affects approximation efficiency, yet numerical comparisons between constructions are not given. The method does not directly apply to non-Gaussian priors or nonlinear observation operators. Moreover, the convergence in Theorem 3.3 holds outside a PY-null set that depends on the chosen version, which practitioners should keep in mind.