On the Relationship between Graphical Gaussian Processes and Functional Gaussian Graphical Models

Multivariate functional or spatial data are commonly analysed using multivariate Gaussian processes (GP). Two recent approaches extend Gaussian graphical models (GGM) to multivariate Gaussian processes with each node representing an entire GP on a continuous domain and edges representing process-level conditional independence. The functional GGM uses a basis function expansion for a multivariate process and models the vector coefficients of the basis functions using GGM. Graphical GP directly constructs valid multivariate covariance functions by "stitching" univariate GPs, while exactly conforming to a given graphical model and retaining marginal properties. We bridge the two seemingly disconnected paradigms by proving that any partially separable functional GGM with a specific covariance selection model is a graphical GP. We also show that the finite term truncation of functional GGM is equivalent to stitching using a low-rank GP, which are known to oversmooth marginal distributions. The theoretical connections help design new algorithms for functional data that exactly conforms to a given inter-variable graph and also retains marginal distributions.