Lattice conditional independence models and Hibi ideals

Peter Caines, Fatemeh Mohammadi, Eduardo Sáenz-de-Cabezón and Henry Wynn

LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library

Abstract: Lattice conditional independence models [Andersson and Perlman, Lattice models for conditional independence in a multivariate normal distribution, Ann. Statist. 21 (1993), 1318–1358] are a class of models developed first for the Gaussian case in which a distributive lattice classifies all the conditional independence statements. The main result is that these models can equivalently be described via a transitive directed acyclic graph (TDAG) in which, as is normal for causal models, the conditional independence is in terms of conditioning on ancestors in the graph. We demonstrate that a parallel stream of research in algebra, the theory of Hibi ideals, not only maps directly to the lattice conditional independence models but gives a vehicle to generalise the theory from the linear Gaussian case. Given a distributive lattice (i) each conditional independence statement is associated with a Hibi relation defined on the lattice, (ii) the directed graph is given by chains in the lattice which correspond to chains of conditional independence, (iii) the elimination ideal of product terms in the chains gives the Hibi ideal and (iv) the TDAG can be recovered from a special bipartite graph constructed via the Alexander dual of the Hibi ideal. It is briefly demonstrated that there are natural applications to statistical log-linear models, time series and Shannon information flow.

JEL-codes: C1 (search for similar items in EconPapers)
Pages: 19 pages
Date: 2022-12-01
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Published in Transactions of the London Mathematical Society, 1, December, 2022, 9(1), pp. 1 - 19. ISSN: 2052-4986

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