Uniform inference in high-dimensional Gaussian graphical models.

The article focuses on developing methodology and theory for uniform inference in high-dimensional Gaussian graphical models, where the number of target parameters can exceed the sample size and approximate sparsity is assumed. It introduces a debiased machine learning approach using nodewise regressions combined with lasso-type estimators, including the square-root lasso, to construct simultaneous confidence regions and conduct joint hypothesis testing on many edges of the precision matrix without relying on strict sparsity or conservative multiple testing corrections. Theoretical results establish uniform convergence rates and sparsity guarantees for nuisance parameter estimation under random design, enabling valid inference on large sets of parameters. Simulation studies demonstrate that the proposed method achieves nominal coverage and improved power compared to existing approaches, and empirical applications to S&P 500 stock data and gene expression data illustrate its practical utility in uncovering conditional dependence structures.