High-Dimensional Mixed Graphical Models

ABSTRACT

While graphical models for continuous data (Gaussian graphical models) and discrete data (Ising models) have been extensively studied, there is little work on graphical models for datasets with both continuous and discrete variables (mixed data), which are common in many scientific applications. We propose a novel graphical model for mixed data, which is simple enough to be suitable for high-dimensional data, yet flexible enough to represent all possible graph structures. We develop a computationally efficient regression-based algorithm for fitting the model by focusing on the conditional log-likelihood of each variable given the rest. The parameters have a natural group structure, and sparsity in the fitted graph is attained by incorporating a group lasso penalty, approximated by a weighted lasso penalty for computational efficiency. We demonstrate the effectiveness of our method through an extensive simulation study and apply it to a music annotation dataset (CAL500), obtaining a sparse and interpretable graphical model relating the continuous features of the audio signal to binary variables such as genre, emotions, and usage associated with particular songs. While we focus on binary discrete variables for the main presentation, we also show that the proposed methodology can be easily extended to general discrete variables.

KEYWORDS:

• Conditional Gaussian density
• Graphical model
• Group lasso
• Mixed variables
• Music annotation

Acknowledgments

The authors thank the associate editor and two referees for their careful reading of the article and many helpful suggestions. This work was performed when the first author was a PhD student at the University of Michigan. E. Levina's research was partially supported by NSF grants DMS-1106772, DMS-1159005, DMS-1521551; J. Zhu's research was partially supported by NSF grant DMS-1407698 and NIH grant R01GM096194.