Conditioning of Banach Space Valued Gaussian Random Variables: An Approximation Approach Based on Martingales
This work provides theoretical foundations for conditioning Gaussian processes in machine learning, offering a rigorous approximation approach that is incremental but improves understanding in functional analysis contexts.
The paper tackles the problem of approximating conditional distributions of jointly Gaussian random variables in Banach spaces, showing that these distributions are Gaussian and can be approximated via a finite-dimensional scheme with convergence in nuclear norm and weak convergence of probabilities. It applies these results to continuous Gaussian processes, demonstrating that conditioning on increasing finite observations yields consistent approximations with uniform convergence of mean and covariance functions.
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.