  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 linking 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 $\ell_1$ penalty for computational
efficiency. We demonstrate the effectiveness of our method through an extensive
simulation study and apply it to a music annotation data set (CAL500),
obtaining a sparse and interpretable graphical model relating the continuous
features of the audio signal to categorical variables such as genre, emotions,
and usage associated with particular songs. While we focus on binary discrete
variables, we also show that the proposed methodology can be easily extended to
general discrete variables.
