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Uniform Inference in High-Dimensional Gaussian Graphical Models

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  • Sven Klaassen
  • Jannis Kuck
  • Martin Spindler
  • Victor Chernozhukov

Abstract

Graphical models have become a very popular tool for representing dependencies within a large set of variables and are key for representing causal structures. We provide results for uniform inference on high-dimensional graphical models with the number of target parameters $d$ being possible much larger than sample size. This is in particular important when certain features or structures of a causal model should be recovered. Our results highlight how in high-dimensional settings graphical models can be estimated and recovered with modern machine learning methods in complex data sets. To construct simultaneous confidence regions on many target parameters, sufficiently fast estimation rates of the nuisance functions are crucial. In this context, we establish uniform estimation rates and sparsity guarantees of the square-root estimator in a random design under approximate sparsity conditions that might be of independent interest for related problems in high-dimensions. We also demonstrate in a comprehensive simulation study that our procedure has good small sample properties.

Suggested Citation

  • Sven Klaassen & Jannis Kuck & Martin Spindler & Victor Chernozhukov, 2018. "Uniform Inference in High-Dimensional Gaussian Graphical Models," Papers 1808.10532, arXiv.org, revised Dec 2018.
  • Handle: RePEc:arx:papers:1808.10532
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    References listed on IDEAS

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    1. Victor Chernozhukov & Denis Chetverikov & Mert Demirer & Esther Duflo & Christian Hansen & Whitney Newey & James Robins, 2016. "Double/Debiased Machine Learning for Treatment and Causal Parameters," Papers 1608.00060, arXiv.org, revised Dec 2017.
    2. Victor Chernozhukov & Denis Chetverikov & Mert Demirer & Esther Duflo & Christian Hansen & Whitney Newey & James Robins, 2018. "Double/debiased machine learning for treatment and structural parameters," Econometrics Journal, Royal Economic Society, vol. 21(1), pages 1-68, February.
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    4. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors," Papers 1212.6906, arXiv.org, revised Jan 2018.
    5. Victor Chernozhukov & Denis Chetverikov & Mert Demirer & Esther Duflo & Christian Hansen & Whitney Newey, 2017. "Double/Debiased/Neyman Machine Learning of Treatment Effects," American Economic Review, American Economic Association, vol. 107(5), pages 261-265, May.
    6. Ming Yuan & Yi Lin, 2007. "Model selection and estimation in the Gaussian graphical model," Biometrika, Biometrika Trust, vol. 94(1), pages 19-35.
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