Bayesian graphical modeling provides an appealing way to obtain uncertaintyestimates when inferring network structures, and much recent progress has beenmade for Gaussian models. These models have been used extensively inapplications to gene expression data, even in cases where there appears to besignificant deviations from the Gaussian model. For more robust inferences, itis natural to consider extensions to t-distribution models. We argue that theclassical multivariate t-distribution, defined using a single latent Gammarandom variable to rescale a Gaussian random vector, is of little use in highlymultivariate settings, and propose other, more flexible t-distributions. Usingan independent Gamma-divisor for each component of the random vector defineswhat we term the alternative t-distribution. The associated model allows one toextract information from highly multivariate data even when most experimentscontain outliers for some of their measurements. However, the use of thisalternative model comes at increased computational cost and imposes constraintson the achievable correlation structures, raising the need for a compromisebetween the classical and alternative models. To this end we propose the use ofDirichlet processes for adaptive clustering of the latent Gamma-scalars, eachof which may then divide a group of latent Gaussian variables. Dirichletprocesses are commonly used to cluster independent observations; here they areused instead to cluster the dependent components of a single observation. Theresulting Dirichlet t-distribution interpolates naturally between the twoextreme cases of the classical and alternative t-distributions and combinesmore appealing modeling of the multivariate dependence structure with favorablecomputational properties.
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