On the non-local priors for sparsity selection in high-dimensional Gaussian DAG models

Abstract

We consider sparsity selection for the Cholesky factor L of the inverse covariance matrix in high-dimensional Gaussian DAG models. The sparsity is induced over the space of L via non-local priors, namely the product moment (pMOM) prior [Johnson, V., & Rossell, D. (2012). Bayesian model selection in high-dimensional settings. Journal of the American Statistical Association, 107(498), 649–660. https://doi.org/10.1080/01621459.2012.682536] and the hierarchical hyper-pMOM prior [Cao, X., Khare, K., & Ghosh, M. (2020). High-dimensional posterior consistency for hierarchical non-local priors in regression. Bayesian Analysis, 15(1), 241–262. https://doi.org/10.1214/19-BA1154]. We establish model selection consistency for Cholesky factor under more relaxed conditions compared to those in the literature and implement an efficient MCMC algorithm for parallel selecting the sparsity pattern for each column of L. We demonstrate the validity of our theoretical results via numerical simulations, and also use further simulations to demonstrate that our sparsity selection approach is competitive with existing methods.

Keywords:

• Bayesian DAG models
• non-local priors
• high-dimensional data
• posterior consistency
• graph selection

Acknowledgments

We would like to thank the Editor, the Associate Editor and the reviewer for their insightful comments which have led to improvements of an earlier version of this paper.

Additional information

Funding

This work was supported by Simons Foundation's collaboration grant (No. 635213).