Lattices of Graphical Gaussian Models with Symmetries

Abstract

In order to make graphical Gaussian models a viable modelling tool when the number of variables outgrows the number of observations, [1] introduced model classes which place equality restrictions on concentrations or partial correlations. The models can be represented by vertex and edge coloured graphs. The need for model selection methods makes it imperative to understand the structure of model classes. We identify four model classes that form complete lattices of models with respect to model inclusion, which qualifies them for an Edwards-Havránek model selection procedure [2]. Two classes turn out most suitable for a corresponding model search. We obtain an explicit search algorithm for one of them and provide a model search example for the other.

Estimation of means in graphical Gaussian models with symmetries

“…This model has previously been found to be well supported by the data [Gehrmann (2011), Højsgaard and Lauritzen (2008), Whittaker (1990)] when no constraints were considered on the means. We may, for example, be interested in the hypothesis that the two lengths have the same mean, and the two breadths have the same mean, indicating that head dimensions do not generally change with the parity of the son.…”

Fused Graphical Lasso for Brain Networks with Symmetries

“…Although the theory of estimation and testing for RCON models is well-established, a procedure that performs model selection within the family of RCON models is not available, with the relevant exception of the procedures introduced by Gehrmann (2011) and by Li et al (2020), within the frequentist and the Bayesian approach, respectively, which are of theoretical interest but whose computational complexity restricts their application to low-dimensional settings. More specifically, the problem of model selection for RCON model is discussed in Gehrmann (2011) where it is shown that the number of RCON models grows super-exponentially in the number of variables. For this reason, Gehrmann (2011) suggested that lasso procedures with fused type penalties might represent a useful alternative to traditional model selection approaches.…”

Model selection in the space of Gaussian models invariant by symmetry

“…This is due in great part to a poor combinatorial description of the colored spaces Z Γ . In particular, the number of such spaces, that is, # { Z Γ ; Γ ∈ S p } is generally unknown for large p. It was shown in Gehrmann (2011) that these colorings constitute a lattice with respect to the usual inclusion of subspaces. However the structure of this lattice is rather complicated and is unobtainable for big p. This, in turn, does not allow to define a Markov chain with known transition probabilities on such colorings.…”

“…Our numbering of colored models on four vertices is in accordance with (Gehrmann, 2011, Figures 15 and 16, p 674-675). However, we identify models by the largest group with the same coloring Γ * rather than the smallest as in Gehrmann (2011). There are 30 different subgroups of S 4 , which generate 22 different colored spaces.…”