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Algebraic properties of {HTC}-identifiable graphs

Bohao Yao and Robin Evans

Vol. 13 (2022), No. 1, 19–39

HTC-identifiable graphs are a large family of graphs known to be generically identifiable. We explore some algebraic properties of linear structural equation models that can be represented by an HTC-identifiable graph. In particular, we prove that all mixed graphs are HTC-identifiable if and only if all the regression coefficients can be recovered from the covariance matrix using straightforward linear algebra operations on specified equations. Furthermore, given an HTC-identifiable graph, we provide a set of polynomials that generates the ideal that encompasses all the equality constraints of the corresponding graphical model on the cone of positive definite matrices. We further prove that, for a subset of HTC-identifiable graphs, this set of polynomials are the minimal generators of said ideal.

algebraic statistics, graphical model, structural equation model, covariance matrix, Gaussian distribution, constraints, identifiability
Mathematical Subject Classification
Primary: 62r01
Received: 23 November 2021
Revised: 3 May 2022
Accepted: 7 June 2022
Published: 4 December 2022
Bohao Yao
Department of Statistics
University of Oxford
United Kingdom
Robin Evans
Department of Statistics
University of Oxford
United Kingdom