Disintegration of Gaussian Measures for Sequential Assimilation of Linear Operator Data

TL;DR

This paper develops a mathematical framework for Gaussian process modeling with linear operator data, extending existing theories to sequential settings and infinite-dimensional spaces, with implications for model updating and posterior analysis.

Contribution

It provides conditions for Gaussian process modeling via Gaussian measures in Banach spaces and extends measure disintegration and update formulas to infinite-dimensional, sequential contexts.

Findings

Established conditions for Gaussian process and measure correspondence in Banach spaces.

Derived discretization-independent update formulas for Gaussian measures in sequential settings.

Extended path property results of GPs to broader infinite-dimensional frameworks.

Abstract

Gaussian processes appear as building blocks in various stochastic models and have been found instrumental to account for imprecisely known, latent functions. It is often the case that such functions may be directly or indirectly evaluated, be it in static or in sequential settings. Here we focus on situations where, rather than pointwise evaluations, evaluations of prescribed linear operators at the function of interest are (sequentially) assimilated. While working with operator data is increasingly encountered in the practice of Gaussian process modelling, mathematical details of conditioning and model updating in such settings are typically by-passed. Here we address these questions by highlighting conditions under which Gaussian process modelling coincides with endowing separable Banach spaces of functions with Gaussian measures, and by leveraging existing results on the disintegration of such measures with respect to operator data. Using recent results on path properties of GPs and their connection to RKHS, we extend the Gaussian process - Gaussian measure correspondence beyond the standard setting of Gaussian random elements in the Banach space of continuous functions. Turning then to the sequential settings, we revisit update formulae in the Gaussian measure framework and establish equalities between final and intermediate posterior mean functions and covariance operators. The latter equalities appear as infinite-dimensional and discretization-independent analogues of Gaussian vector update formulae.