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Algebraic properties of {HTC}-identifiable graphs

Bohao Yao and Robin Evans

Vol. 13 (2022), No. 1, 19–39
Abstract

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.

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