A matrix algebra for graphical statistical models

TL;DR

This paper develops a matrix algebra framework for directed mixed graphs, enabling formalization and visualization of complex graphical and probabilistic concepts in statistical models, especially Gaussian systems.

Contribution

It introduces a novel matrix algebra for walks on directed mixed graphs, linking graphical concepts with probabilistic properties in Gaussian models.

Findings

Formalizes graphical concepts like latent projection and separation

Connects graphical structures with probabilistic independence

Enables visualization of complex graphical proofs

Abstract

Directed mixed graphs permit directed and bidirected edges between any two vertices. They were first considered in the path analysis developed by Sewall Wright and play an essential role in statistical modeling. We introduce a matrix algebra for walks on such graphs. Each element of the algebra is a matrix whose entries are sets of walks on the graph from the corresponding row to the corresponding column. The matrix algebra is then generated by applying addition (set union), multiplication (concatenation), and transpose to the two basic matrices consisting of directed and bidirected edges. We use it to formalize, in the context of Gaussian linear systems, the correspondence between important graphical concepts such as latent projection and graph separation with important probabilistic concepts such as marginalization and (conditional) independence. In two further examples regarding confounder adjustment and the augmentation criterion, we illustrate how the algebra allows us to visualize complex graphical proofs. A "dictionary" and LATEX macros for the matrix algebra are provided in the Appendix.