One-connection rule for structural equation models

TL;DR

This paper extends algebraic understanding of linear structural equation models to cyclic graphs by providing a closed form for covariance matrices and analyzing the model's identifiability.

Contribution

It introduces a closed form expression for covariance matrices in models with cycles and shows the parametrization is generically finite-to-one for simple graphs.

Findings

Closed form expression for covariance matrices in cyclic models.

The parametrization map is generically finite-to-one for simple graphs.

Enables systematic exploration of polynomials in the Gaussian vanishing ideal.

Abstract

Linear structural equation models are multivariate statistical models encoded by mixed graphs. In particular, the set of covariance matrices for distributions belonging to a linear structural equation model for a fixed mixed graph $G=(V, D,B)$ is parameterized by a rational function with parameters for each vertex and edge in $G$. This rational parametrization naturally allows for the study of these models from an algebraic and combinatorial point of view. Indeed, this point of view has led to a collection of results in the literature, mainly focusing on questions related to identifiability and determining relationships between covariances (i.e., finding polynomials in the Gaussian vanishing ideal). So far, a large proportion of these results has focused on the case when $D$, the directed part of the mixed graph $G$, is acyclic. This is due to the fact that in the acyclic case, the parametrization becomes polynomial and there is a description of the entries of the covariance matrices in terms of a finite sum. We move beyond the acyclic case and give a closed form expression for the entries of the covariance matrices in terms of the one-connections in a graph obtained from $D$ through some small operations. This closed form expression then allows us to show that if $G$ is simple, then the parametrization map is generically finite-to-one. Finally, having a closed form expression for the covariance matrices allows for the development of an algorithm for systematically exploring possible polynomials in the Gaussian vanishing ideal.