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

Joint Mean-Covariance Estimation via the Horseshoe with an Application in Genomic Data Analysis