An Expectation Conditional Maximization Approach for Gaussian Graphical Models

Abstract

Bayesian graphical models are a useful tool for understanding dependence relationships among many variables, particularly in situations with external prior information. In high-dimensional settings, the space of possible graphs becomes enormous, rendering even state-of-the-art Bayesian stochastic search computationally infeasible. We propose a deterministic alternative to estimate Gaussian and Gaussian copula graphical models using an expectation conditional maximization (ECM) algorithm, extending the EM approach from Bayesian variable selection to graphical model estimation. We show that the ECM approach enables fast posterior exploration under a sequence of mixture priors, and can incorporate multiple sources of information. Supplementary materials for this article are available online.

Keywords:

• Copula graphical model
• Sparse precision matrix
• Spike-and-slab prior

Acknowledgments

We would like to thank Jon Wakefield, Sam Clark, Johannes Lederer, Adrian Dobra, Daniela Witten, and Matt Taddy for helpful discussions and feedback.

Notes

1 ¹ http://www.seattle.gov/tech/initiatives/open-data/

Additional information

Funding

The authors gratefully acknowledge grants SES-1559778 and DMS-1737673 from the National Science foundation, and grants from the National Institute of Child Health and Human Development (NICHD), NIH funding: K01 HD078452 and R01 HD086227.