  "Mixed Data" comprising a large number of heterogeneous variables (e.g.
count, binary, continuous, skewed continuous, among other data types) are
prevalent in varied areas such as genomics and proteomics, imaging genetics,
national security, social networking, and Internet advertising. There have been
limited efforts at statistically modeling such mixed data jointly, in part
because of the lack of computationally amenable multivariate distributions that
can capture direct dependencies between such mixed variables of different
types. In this paper, we address this by introducing a novel class of Block
Directed Markov Random Fields (BDMRFs). Using the basic building block of
node-conditional univariate exponential families from Yang et al. (2012), we
introduce a class of mixed conditional random field distributions, that are
then chained according to a block-directed acyclic graph to form our class of
Block Directed Markov Random Fields (BDMRFs). The Markov independence graph
structure underlying a BDMRF thus has both directed and undirected edges. We
introduce conditions under which these distributions exist and are
normalizable, study several instances of our models, and propose scalable
penalized conditional likelihood estimators with statistical guarantees for
recovering the underlying network structure. Simulations as well as an
application to learning mixed genomic networks from next generation sequencing
expression data and mutation data demonstrate the versatility of our methods.
