  Inferring dependence structure through undirected graphs is crucial for
uncovering the major modes of multivariate interaction among high-dimensional
genomic markers that are potentially associated with cancer. Traditionally,
conditional independence has been studied using sparse Gaussian graphical
models for continuous data and sparse Ising models for discrete data. However,
there are two clear situations when these approaches are inadequate. The first
occurs when the data are continuous but display non-normal marginal behavior
such as heavy tails or skewness, rendering an assumption of normality
inappropriate. The second occurs when a part of the data is ordinal or discrete
(e.g., presence or absence of a mutation) and the other part is continuous
(e.g., expression levels of genes or proteins). In this case, the existing
Bayesian approaches typically employ a latent variable framework for the
discrete part that precludes inferring conditional independence among the data
that are actually observed. The current article overcomes these two challenges
in a unified framework using Gaussian scale mixtures. Our framework is able to
handle continuous data that are not normal and data that are of mixed
continuous and discrete nature, while still being able to infer a sparse
conditional sign independence structure among the observed data. Extensive
performance comparison in simulations with alternative techniques and an
analysis of a real cancer genomics data set demonstrate the effectiveness of
the proposed approach.
