Loss function, unbiasedness, and optimality of Gaussian graphical model selection

• Published in: Journal of statistical planning and inference Vol. 201; pp. 32 - 39
• Main Authors: Kalyagin, Valery A., Koldanov, Alexander P., Koldanov, Petr A., Pardalos, Panos M.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.07.2019
• Subjects: Gaussian graphical model; Loss function; Optimal multiple decision statistical procedures; Risk function; Tests of a Neyman structure; Unbiased multiple decision statistical procedures; Loss function; Unbiased multiple decision statistical procedures; 05C82; 90B15; Optimal multiple decision statistical procedures; Risk function; Gaussian graphical model; Tests of a Neyman structure
• ISSN: 0378-3758; 1873-1171
• DOI: 10.1016/j.jspi.2018.11.002

A Gaussian graphical model is a graphical representation of the dependence structure for a Gaussian random vector. Gaussian graphical model selection is a statistical problem that identifies the Gaussian graphical model from observations. There are several statistical approaches for Gaussian graphical model identification. Their properties, such as unbiasedeness and optimality, are not established. In this paper we study these properties. We consider the graphical model selection problem in the framework of multiple decision theory and suggest assessing these procedures using an additive loss function. Associated risk function in this case is a linear combination of the expected numbers of the two types of error (False Positive and False Negative). We combine the tests of a Neyman structure for individual hypotheses with simultaneous inference and prove that the obtained multiple decision procedure is optimal in the class of unbiased multiple decision procedures. •General approach to measure uncertainty of Gaussian graphical model selection procedures.•Proof of the optimality of the procedure of simultaneous inference for Gaussian graphical model selection.•Discussion of the unbiasedness in multiple testing.