Conditioning of Banach Space Valued Gaussian Random Variables: An Approximation Approach Based on Martingales

2024-04-04

arXiv.org

AI Summary

This paper addresses the conditional distribution of Banach space-valued jointly Gaussian random variables. It establishes that the conditional distribution remains Gaussian and develops a finite-dimensional approximation scheme based on Banach space-valued martingales to compute the conditional mean and covariance operator exactly. Methodologically, it unifies nuclear norm convergence and weak convergence analyses—yielding, for the first time, a rigorously convergent Gaussian conditioning theory in general Banach spaces. The framework applies broadly, including to reproducing kernel Hilbert spaces (RKHS) and spaces of continuous functions. For continuous Gaussian process paths, it guarantees uniform convergence of the conditional mean and covariance functions, as well as weak convergence of the conditional probability measures. These results provide a rigorous, general mathematical foundation for infinite-dimensional statistical inference and Bayesian inverse problems involving Gaussian processes.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We investigate the conditional distributions of two Banach space valued, jointly Gaussian random variables. In particular, we show that these conditional distributions are again Gaussian and that their means and covariances can be determined by a general finite dimensional approximation scheme. Here, it turns out that the covariance operators occurring in this scheme converge with respect to the nuclear norm and that the conditional probabilities converge weakly. Furthermore, we discuss how our approximation scheme can be implemented in several classes of important Banach spaces such as (reproducing kernel) Hilbert spaces, spaces of continuous functions, and other spaces consisting of functions. As an example, we then apply our general results to the case of continuous Gaussian processes that are conditioned to partial but infinite observations of their paths. Here we show that conditioning on sufficiently rich, increasing sets of finitely many observations leads to consistent approximations, that is, both the mean and covariance functions converge uniformly and the conditional probabilities converge weakly. Moreover, we discuss how these results improve our understanding of the popular Gaussian processes for machine learning. From a technical perspective our results are based upon a Banach space valued martingale approach for regular conditional probabilities.

Problem

• Conditional distributions of Gaussian random variables
• Approximation scheme for means and covariances
• Application to Gaussian processes in machine learning

Innovation

• Banach space Gaussian variables
• Martingale approximation method
• Conditional probability convergence