Hierarchical Normalized Completely Random Measures for Robust Graphical Modeling

Abstract. Gaussian graphical models are useful tools for exploring network structures in multivariate normal data. In this paper we are interested in situations\r\nwhere data show departures from Gaussianity, therefore requiring alternative modeling distributions. The multivariate t-distribution, obtained by dividing each component of the data vector by a gamma random variable, is a straightforward generalization to accommodate deviations from normality such as heavy tails. Since\r\ndifferent groups of variables may be contaminated to a different extent, Finegold\r\nand Drton (2014) introduced the Dirichlet t-distribution, where the divisors are\r\nclustered using a Dirichlet process. In this work, we consider a more general class\r\nof nonparametric distributions as the prior on the divisor terms, namely the class\r\nof normalized completely random measures (NormCRMs). To improve the effectiveness of the clustering, we propose modeling the dependence among the divisors\r\nthrough a nonparametric hierarchical structure, which allows for the sharing of\r\nparameters across the samples in the data set. This desirable feature enables us\r\nto cluster together different components of multivariate data in a parsimonious\r\nway. We demonstrate through simulations that this approach provides accurate\r\ngraphical model inference, and apply it to a case study examining the dependence\r\nstructure in radiomics data derived from The Cancer Imaging Atlas.