Optimal statistical decision for Gaussian graphical model selection

TL;DR

This paper develops an optimal statistical decision procedure for Gaussian graphical model selection that balances Type I and Type II errors using multiple decision theory, improving accuracy in identifying dependence structures.

Contribution

It introduces a new optimal multiple decision framework for Gaussian graphical model selection that accounts for both types of errors, unlike existing methods focused mainly on Type I error control.

Findings

The proposed procedure minimizes a linear combination of expected Type I and Type II errors.

The method constructs Neyman-structured tests for individual hypotheses and combines them optimally.

The approach is proven to be optimal among unbiased multiple decision procedures.

Abstract

Gaussian graphical model is a graphical representation of the dependence structure for a Gaussian random vector. It is recognized as a powerful tool in different applied fields such as bioinformatics, error-control codes, speech language, information retrieval and others. Gaussian graphical model selection is a statistical problem to identify the Gaussian graphical model from a sample of a given size. Different approaches for Gaussian graphical model selection are suggested in the literature. One of them is based on considering the family of individual conditional independence tests. The application of this approach leads to the construction of a variety of multiple testing statistical procedures for Gaussian graphical model selection. An important characteristic of these procedures is its error rate for a given sample size. In existing literature great attention is paid to the control of error rates for incorrect edge inclusion (Type I error). However, in graphical model selection it is also important to take into account error rates for incorrect edge exclusion (Type II error). To deal with this issue we consider the graphical model selection problem in the framework of the multiple decision theory. The quality of statistical procedures is measured by a risk function with additive losses. Additive losses allow both types of errors to be taken into account. We construct the tests of a Neyman structure for individual hypotheses and combine them to obtain a multiple decision statistical procedure. We show that the obtained procedure is optimal in the sense that it minimizes the linear combination of expected numbers of Type I and Type II errors in the class of unbiased multiple decision procedures.