  Seemingly unrelated regression is a natural framework for regressing multiple
correlated responses on multiple predictors. The model is very flexible, with
multiple linear regression and covariance selection models being special cases.
However, its practical deployment in genomic data analysis under a Bayesian
framework is limited due to both statistical and computational challenges. The
statistical challenge is that one needs to infer both the mean vector and the
inverse covariance matrix, a problem inherently more complex than separately
estimating each. The computational challenge is due to the dimensionality of
the parameter space that routinely exceeds the sample size. We propose the use
of horseshoe priors on both the mean vector and the inverse covariance matrix.
This prior has demonstrated excellent performance when estimating a mean vector
or inverse covariance matrix separately. The current work shows these
advantages are also present when addressing both simultaneously. A full
Bayesian treatment is proposed, with a sampling algorithm that is linear in the
number of predictors. MATLAB code implementing the algorithm is freely
available from github at https://github.com/liyf1988/HS_GHS. Extensive
performance comparisons are provided with both frequentist and Bayesian
alternatives, and both estimation and prediction performances are verified on a
genomic data set.
