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We consider the problem of jointly estimating a collection of graphicalmodels for discrete data, corresponding to several categories that share somecommon structure. An example for such a setting is voting records oflegislators on different issues, such as defense, energy, and healthcare. Wedevelop a Markov graphical model to characterize the heterogeneous dependencestructures arising from such data. The model is fitted via a joint estimationmethod that preserves the underlying common graph structure, but also allowsfor differences between the networks. The method employs a group penalty thattargets the common zero interaction effects across all the networks. We applythe method to describe the internal networks of the U.S. Senate on severalimportant issues. Our analysis reveals individual structure for each issue,distinct from the underlying well-known bipartisan structure common to allcategories which we are able to extract separately. We also establishconsistency of the proposed method both for parameter estimation and modelselection, and evaluate its numerical performance on a number of simulatedexamples.

Estimating heterogeneous graphical models for discrete data with an application to roll call voting