Quasi‐Symmetric Graphical Log‐Linear Models

Abstract

ABSTRACT. We propose an extension of graphical log-linear models to allow for symmetry constraints on some interaction parameters that represent homologous factors. The conditional independence structure of such quasi-symmetric (QS) graphical models is described by an undirected graph with coloured edges, in which a particular colour corresponds to a set of equality constraints on a set of parameters. Unlike standard QS models, the proposed models apply with contingency tables for which only some variables or sets of the variables have the same categories. We study the graphical properties of such models, including conditions for decomposition of model parameters and of maximum likelihood estimates.

Lattices of Graphical Gaussian Models with Symmetries

“…The graph colourings in K V which are generated by a single permutation, i.e., by Γ = σ for σ ∈ S(V ), are displayed in Figure 15, where, for the sake of legibility we label the vertices by V = {1, 2, 3, 4}. The remaining 13 subgroups generate only 5 distinct colourings in K [4] , all of which are shown in Figure 16 with one of their generating groups. Figure 16.…”

“…The graphs displayed in Figure 10 establish non-distributivity of Π V as each of them has a permutation-generated colouring, with Γ 6 = (124) , Γ 7 = (234) , Γ 8 = (13), (24) , Γ 7∧8 = S( [4]) and Γ 6∨7 = Γ 6∨8 = (24) respectively. The above directly translate to S + Π V , proving the claim.…”

“…Figure 16. Remaining colourings in K [4] . r r r r rë e e e e e e e generating groups are given in Figure 18.…”

“…Figure 9. B [4] is not stable under ∨. Let G = (V, E) ∈ C V and let E B be the partition of E which puts αβ, γδ ∈ E in the same set whenever they connect the same vertex color classes.…”

“…Motivated by the growing need for parsimonious models in modern day applications, in particular when the number of variables outgrows the number of observations, in recent years graphical models with additional equality constraints on model parameters are becoming of increasing interest, in discrete models [4,5] as well as in multivariate Gaussian models, which are the central object of interest in this article. First studies [1,6] show that equality constraints reduce the minimal number of observations required to ensure estimability of the model parameters with probability one, which makes graphical Gaussian models with equality constraints a promising model class.…”