Local and Global Inference for High Dimensional Nonparanormal Graphical Models

Abstract

This paper proposes a unified framework to quantify local and global inferential uncertainty for high dimensional nonparanormal graphical models. In particular, we consider the problems of testing the presence of a single edge and constructing a uniform confidence subgraph. Due to the presence of unknown marginal transformations, we propose a pseudo likelihood based inferential approach. In sharp contrast to the existing high dimensional score test method, our method is free of tuning parameters given an initial estimator, and extends the scope of the existing likelihood based inferential framework. Furthermore, we propose a U-statistic multiplier bootstrap method to construct the confidence subgraph. We show that the constructed subgraph is contained in the true graph with probability greater than a given nominal level. Compared with existing methods for constructing confidence subgraphs, our method does not rely on Gaussian or sub-Gaussian assumptions. The theoretical properties of the proposed inferential methods are verified by thorough numerical experiments and real data analysis.

A Unified Theory of Confidence Regions and Testing for High-Dimensional Estimating Equations

“…There are a number of recent works considering the inferential problems for Gaussian graphical models (Jankova and van de Geer, 2014;Chen et al, 2015;Ren et al, 2015;Liu, 2013) and Gaussian copula graphical models (Gu et al, 2015;Barber and Kolar, 2015). Our framework differs from these existing procedures in the following two aspects.…”

Combinatorial inference for graphical models

“…A central theme of graphical model research is to infer the structure of the underlying graph based on observational data. Though significant progress has been made, existing works mainly focus on estimating the graph (Meinshausen and Bühlmann, 2006;Liu et al, 2009;Ravikumar et al, 2011;Cai et al, 2011) or testing the existence of a single edge (Jankova and van de Geer, 2015;Ren et al, 2015;Neykov et al, 2015;Gu et al, 2015).…”

“…Recently, motivated by Zhang and Zhang (2014), various inferential methods for high-dimensional graphical models were suggested (Liu, 2013;Jankova and van de Geer, 2015;Chen et al, 2015;Ren et al, 2015;Neykov et al, 2015;Gu et al, 2015, e.g. ), most of which focus on testing the presence of a single edge (except Liu (2013) who took the FDR approach (Benjamini and Hochberg, 1995) to conduct multiple tests and Gu et al (2015) who developed procedures of edge testing in Gaussian copula models). None of the aforementioned works address the problem of combinatorial structure testing.…”

High-Temperature Structure Detection in Ferromagnets

“…The analysis of graph structures plays a fundamental role in a wide variety of applications, including information retrieval, bioinformatics, image processing and social networks (Besag, 1993;Durbin et al, 1998;Wasserman and Faust, 1994;Grabowski and Kosiński, 2006). Motivated by these applications, theoretical results on graph estimation (Meinshausen and Bühlmann, 2006;Liu et al, 2009;Montanari and Pereira, 2009;Ravikumar et al, 2011;Cai et al, 2011), single edge inference (Jankova et al, 2015;Ren et al, 2015;Neykov et al, 2015;Gu et al, 2015) and combinatorial inference (Neykov et al, 2016;Neykov and Liu, 2017) have been studied in the literature.…”