[Paper Review] A SINful Approach to Gaussian Graphical Model Selection

Mathias Drton, Michael D. Perlman|arXiv (Cornell University)|Aug 15, 2005

Bayesian Modeling and Causal Inference10 citations

TL;DR

This paper proposes the SIN method for Gaussian graphical model selection by leveraging Fisher’s z-transformation, Šidák’s inequality, and Holm’s step-down procedure to test conditional independences via partial correlations. It controls the family-wise error rate for edge inclusion and outputs two graphs—one inclusive (S ∪ I) and one conservative (S only)—enabling robust, error-controlled model selection with flexible incorporation of prior edge information.

ABSTRACT

Abstract. Multivariate Gaussian graphical models are defined in terms of Markov properties, i.e., conditional independences associated with the underlying graph. Thus, model selection can be performed by testing these conditional independences, which are equivalent to specified zeroes among certain (partial) correlation coefficients. For concentration graphs, covariance graphs, acyclic directed graphs, and chain graphs (both LWF and AMP), we apply Fisher’s z-transformation, ˇ Sidák’s correlation inequality, and Holm’s step-down procedure, to simultaneously test the multiple hypotheses obtained from the Markov properties. This leads to a simple method for model selection that controls the overall error rate for incorrect edge inclusion. In practice, we advocate partitioning the simultaneous p-values into three disjoint sets, a significant set S, an indeterminate set I, and a non-significant set N. Then our SIN model selection method selects two graphs, a graph whose edges correspond to the union of S and I, and a more conservative graph whose edges correspond to S only. Prior information about the presence and/or absence of particular edges can be incorporated readily. 1.

Motivation & Objective

• To develop a statistically rigorous method for selecting Gaussian graphical models that controls the overall error rate for incorrect edge inclusion.
• To address the challenge of multiple hypothesis testing in graphical model selection by applying established statistical corrections to partial correlation tests.
• To provide a practical framework that distinguishes between significant, indeterminate, and non-significant partial correlations for informed model construction.
• To enable integration of prior knowledge about edge presence or absence into the model selection process.
• To offer two complementary graphical models—one inclusive and one conservative—based on statistical evidence from partial correlations.

Proposed method

• Apply Fisher’s z-transformation to stabilize the sampling distribution of partial correlation coefficients for hypothesis testing.
• Use Šidák’s correlation inequality to adjust p-values and control the family-wise error rate across multiple simultaneous tests.
• Implement Holm’s step-down procedure to sequentially test hypotheses and improve power while maintaining error control.
• Partition p-values into three disjoint sets: significant (S), indeterminate (I), and non-significant (N) based on adjusted thresholds.
• Construct two graphs: one using edges from S ∪ I (inclusive), and another using only edges from S (conservative).
• Incorporate prior information about specific edges by pre-defining their inclusion or exclusion in the hypothesis testing framework.

Experimental results

Research questions

• RQ1 How can multiple conditional independence tests in Gaussian graphical models be simultaneously controlled to prevent false edge inclusion?
• RQ2 What statistical procedure best balances Type I error control and power in high-dimensional partial correlation testing?
• RQ3 How can prior knowledge about graph structure be systematically integrated into a hypothesis-testing framework for model selection?
• RQ4 What is the optimal way to partition p-values from multiple tests into meaningful decision categories for model construction?
• RQ5 Can a two-graph output strategy (inclusive and conservative) improve interpretability and reliability in model selection?

Key findings

• The SIN method effectively controls the family-wise error rate for incorrect edge inclusion through rigorous application of Šidák correction and Holm’s step-down procedure.
• The partitioning of p-values into S, I, and N sets enables a principled approach to model selection with clear interpretability of statistical evidence.
• The inclusive graph (S ∪ I) captures potentially meaningful edges while the conservative graph (S only) minimizes false positives, offering a balanced decision framework.
• The method is generalizable across various types of graphical models, including concentration, covariance, acyclic directed, and chain graphs (LWF and AMP).
• Prior knowledge about specific edges can be seamlessly incorporated into the testing framework without compromising error rate control.