The Graphical Horseshoe Estimator for Inverse Covariance Matrices

Abstract

We develop a new estimator of the inverse covariance matrix for high-dimensional multivariate normal data using the horseshoe prior. The proposed graphical horseshoe estimator has attractive properties compared to other popular estimators, such as the graphical lasso and the graphical smoothly clipped absolute deviation. The most prominent benefit is that when the true inverse covariance matrix is sparse, the graphical horseshoe provides estimates with small information divergence from the sampling model. The posterior mean under the graphical horseshoe prior can also be almost unbiased under certain conditions. In addition to these theoretical results, we also provide a full Gibbs sampler for implementing our estimator. MATLAB code is available for download from github at http://github.com/liyf1988/GHS. The graphical horseshoe estimator compares favorably to existing techniques in simulations and in a human gene network data analysis. Supplementary materials for this article are available online.

Keywords:

• Bayesian shrinkage estimation
• Gaussian graphical model
• High-dimensional graphs
• Sparse precision matrix

Acknowledgments

We thank the AE and two anonymous referees for many constructive suggestions.

Additional information

Funding

The research of Bhadra is partially supported by grant no. DMS-1613063 by the US National Science Foundation.