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Statistical inference using debiased group graphical lasso for multiple sparse precision matrices

Sayan Ranjan Bhowal, Debashis Paul, Gopal K Basak, Samarjit Das

TL;DR

The paper develops a debiased inference framework for the group graphical lasso across multiple Gaussian graphical models sharing a common sparsity pattern. It establishes convergence and model-selection consistency under irrepresentability-like conditions and constructs debiased estimators that are asymptotically Gaussian, enabling hypothesis testing for linear combinations of cross-population precision-matrix entries. The authors extend the theory to settings without irrepresentability, showing consistency in moderately high dimensions, and validate the approach with extensive simulations and three real-world datasets (climatic, WebKb, and weekly returns). The work provides a practical toolkit for cross-population edge-inference in high-dimensional GGMs, including confidence intervals and tests for differences in sparse precision structures.

Abstract

Debiasing group graphical lasso estimates enables statistical inference when multiple Gaussian graphical models share a common sparsity pattern. We analyze the estimation properties of group graphical lasso, establishing convergence rates and model selection consistency under irrepresentability conditions. Based on these results, we construct debiased estimators that are asymptotically Gaussian, allowing hypothesis testing for linear combinations of precision matrix entries across populations. We also investigate regimes where irrepresentibility conditions does not hold, showing that consistency can still be attained in moderately high-dimensional settings. Simulation studies confirm the theoretical results, and applications to real datasets demonstrate the practical utility of the method.

Statistical inference using debiased group graphical lasso for multiple sparse precision matrices

TL;DR

The paper develops a debiased inference framework for the group graphical lasso across multiple Gaussian graphical models sharing a common sparsity pattern. It establishes convergence and model-selection consistency under irrepresentability-like conditions and constructs debiased estimators that are asymptotically Gaussian, enabling hypothesis testing for linear combinations of cross-population precision-matrix entries. The authors extend the theory to settings without irrepresentability, showing consistency in moderately high dimensions, and validate the approach with extensive simulations and three real-world datasets (climatic, WebKb, and weekly returns). The work provides a practical toolkit for cross-population edge-inference in high-dimensional GGMs, including confidence intervals and tests for differences in sparse precision structures.

Abstract

Debiasing group graphical lasso estimates enables statistical inference when multiple Gaussian graphical models share a common sparsity pattern. We analyze the estimation properties of group graphical lasso, establishing convergence rates and model selection consistency under irrepresentability conditions. Based on these results, we construct debiased estimators that are asymptotically Gaussian, allowing hypothesis testing for linear combinations of precision matrix entries across populations. We also investigate regimes where irrepresentibility conditions does not hold, showing that consistency can still be attained in moderately high-dimensional settings. Simulation studies confirm the theoretical results, and applications to real datasets demonstrate the practical utility of the method.

Paper Structure

This paper contains 19 sections, 17 theorems, 132 equations, 24 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Notations
  3. Model setup
  4. Convergence of precision matrices
  5. Results related to debiased estimates of group graphical lasso
  6. Statistical Inference without assuming irrepresentability conditions, between-group irrepresentability conditions, and similar sparsity pattern
  7. Empirical results
  8. Simulation study
  9. Model consistency against sample size
  10. True positives and false positives against sample size
  11. Model consistency against maximum degree
  12. Sup-norm convergence against sample size
  13. Asymptotic normality of the test statistic
  14. Average coverage of the true parameter and average length of the confidence interval
  15. Real-world dataset
  16. ...and 4 more sections

Key Result

Lemma 1.1

Suppose $\boldsymbol{x}^{k},k=1,2,\ldots,K$, are each zero mean random vector with covariance matrix $\boldsymbol{\Sigma}_{0}^{k}$, satisfying the sub-Gaussian conditions (Definition Definition 1.2.2), with a common $K_1>0$. If we have $n_k$ samples each from population model $k, k=1,2,\ldots,K$, th for all $\delta\in (0,\underset{k_1}{min}\ 8(1+12K_1^{2})\ max\ (\Sigma_{0ii}^{k_1}))$.

Figures (24)

  • Figure 1: Star shaped graphs
  • Figure 2: Model consistency against sample size
  • Figure 3: True positive and false positive graphs against sample size (Chain graph)
  • Figure 4: True positive and false positive graphs against sample size (Star graph)
  • Figure 5: Model consistency against maximum degree for star graphs
  • ...and 19 more figures

Theorems & Definitions (38)

  • Definition 1.2.1
  • Definition 1.2.2
  • Definition 1.2.3
  • Definition 1.2.4
  • Lemma 1.1
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 28 more