Conditional Independence in Stationary Diffusions

TL;DR

This paper characterizes the conditional independence relations in stationary distributions of multivariate diffusion processes with sparse drift structures, using a graphical representation to connect drift sparsity with probabilistic independence.

Contribution

It provides a novel characterization of conditional independence in stationary diffusions based on the sparsity pattern of the drift, linking graphical structures to probabilistic relations.

Findings

Conditional independence relations are characterized by the drift's sparsity pattern.

Graphical representations effectively encode the conditional independence structure.

Results apply broadly to stationary multivariate diffusion processes with sparse drifts.

Abstract

Stationary distributions of multivariate diffusion processes have recently been proposed as probabilistic models of causal systems in statistics and machine learning. Motivated by these developments, we study stationary multivariate diffusion processes with a sparsely structured drift. Our main result gives a characterization of the conditional independence relations that hold in a stationary distribution. The result draws on a graphical representation of the drift structure and pertains to conditional independence relations that hold generally as a consequence of the drift's sparsity pattern.