[Paper Review] On perfectness in Gaussian graphical models

Arash A. Amini, Bryon Aragam+1 more|arXiv (Cornell University)|Sep 3, 2019

Bayesian Modeling and Causal Inference13 references4 citations

TL;DR

This paper provides a direct, transparent parametrization of Gaussian graphical models (GGMs) to prove that almost all such models are perfect, meaning conditional independence in the distribution exactly corresponds to graph separation. By extending a construction from Lnička and Matúš, the authors establish strong completeness of GGMs and characterize the null set of imperfect covariance matrices, offering a constructive framework for Monte Carlo sampling over positive definite matrices with graph-restricted support.

ABSTRACT

Knowing when a graphical model is perfect to a distribution is essential in order to relate separation in the graph to conditional independence in the distribution, and this is particularly important when performing inference from data. When the model is perfect, there is a one-to-one correspondence between conditional independence statements in the distribution and separation statements in the graph. Previous work has shown that almost all models based on linear directed acyclic graphs as well as Gaussian chain graphs are perfect, the latter of which subsumes Gaussian graphical models (i.e., the undirected Gaussian models) as a special case. However, the complexity of chain graph models leads to a proof of this result which is indirect and mired by the complications of parameterizing this general class. In this paper, we directly approach the problem of perfectness for the Gaussian graphical models, and provide a new proof, via a more transparent parametrization, that almost all such models are perfect. Our approach is based on, and substantially extends, a construction of Lněnička and Matúš showing the existence of a perfect Gaussian distribution for any graph.

Motivation & Objective

• To establish that almost all Gaussian graphical models (GGMs) are perfect, i.e., conditional independence in the distribution matches separation in the graph.
• To provide a more transparent and direct parametrization of GGMs compared to indirect methods used in chain graph models.
• To characterize the set of imperfect covariance matrices as a null set, offering insight into the genericity of perfectness in GGMs.
• To construct a probability measure over inverse covariance matrices supported on the edge set of a graph, useful for Monte Carlo sampling.

Proposed method

• Extends the constructive method of Lnička and Matúš to fully parametrize the set of GGM-Markovian Gaussian distributions.
• Introduces a novel parametrization using positive definite matrices and edge-specific parameters to ensure Markov compatibility.
• Uses permutation-based cycle decomposition to analyze the structure of covariance matrices and identify conditions under which perfectness fails.
• Applies measure-theoretic arguments to show that the set of imperfect distributions has Lebesgue measure zero, implying almost all GGMs are perfect.
• Employs a diffeomorphic transformation to preserve null sets and prove absolute continuity of the pushforward measure.
• Derives a probability measure on inverse covariance matrices with support restricted to the graph’s edge set, enabling proposal distributions in MCMC algorithms.

Experimental results

Research questions

• RQ1 Are almost all Gaussian graphical models perfect, such that conditional independence in the distribution corresponds exactly to graph separation?
• RQ2 Can a direct and transparent parametrization of GGMs be constructed to simplify the proof of perfectness compared to indirect chain graph methods?
• RQ3 What is the structure of the set of imperfect covariance matrices in GGMs, and can it be characterized as a null set?
• RQ4 Can a probability measure be constructed over inverse covariance matrices that respects the graph’s edge structure for use in simulation?

Key findings

• Almost all Gaussian graphical models are perfect, meaning conditional independence statements in the distribution are in one-to-one correspondence with separation statements in the graph.
• The set of imperfect covariance matrices forms a Lebesgue null set, confirming that perfectness is generic in GGMs.
• A new, direct parametrization of GGM-Markovian distributions is developed, extending the work of Lnička and Matúš.
• The paper constructs a probability measure on inverse covariance matrices supported on the edge set of a graph, enabling use as a proposal distribution in Monte Carlo methods.
• The proof technique provides a constructive description of the null set of imperfect distributions, enhancing intuition for modeling and estimation.
• The measure-theoretic argument confirms that the pushforward of the Lebesgue measure under the parametrization map is absolutely continuous, ensuring generic perfectness.