Structural learning of Gaussian graphical models in the presence of latent variables has long been a challenging problem. Chandrasekaran et al. (2012) proposed a convex program for estimating a sparse graph plus a low-rank term that adjusts for latent variables; however, this approach poses challenges from both computational and statistical perspectives. We propose an alternative, simple solution: apply a hard-thresholding operator to existing graph selection methods. Conceptually simple and computationally attractive, the approach of thresholding the graphical lasso is shown to be graph selection consistent in the presence of latent variables under a simpler minimum edge strength condition and at an improved statistical rate. The results are extended to estimators for thresholded neighbourhood selection and constrained $\ell_{1}$-minimization for inverse matrix estimation as well. We show that our simple thresholded graph estimators yield stronger empirical results than existing methods for the latent variable graphical model problem, and we apply them to a neuroscience case study on estimating functional neural connections.
This paper considers the latent Gaussian graphical model, which extends the Gaussian graphical model to handle discrete data as well as mixed data with both continuous and discrete variables by assuming that discrete variables are generated by discretizing latent Gaussian variables. We propose a modified expectation‐maximization (EM) algorithm to estimate parameters in the latent Gaussian model for binary data. We also extend the proposed modified EM algorithm to the latent Gaussian model for mixed data. The conditional dependence structure can be consequently constructed by exploring the sparsity pattern of the precision matrix of the latent variables. We illustrate the performance of our proposed estimator through comprehensive numerical studies and an application to voting data of the United Nations General Assembly.
more » « less- NSF-PAR ID:
- 10367194
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Canadian Journal of Statistics
- Volume:
- 50
- Issue:
- 2
- ISSN:
- 0319-5724
- Page Range / eLocation ID:
- p. 612-637
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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