Total positivity in structured binary distributions
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We study binary distributions that are multivariate totally positive of order 2 (MTP2). Binary distributions can be represented as an exponential family and we show that MTP2 exponential families are convex. Moreover, MTP2 quadratic exponential families, which contain ferromagnetic Ising models and attractive Gaussian graphical models, are defined by intersecting the space of canonical parameters with a polyhedral cone whose faces correspond to conditional independence relations. Hence MTP2 serves as an implicit regularizer for quadratic exponential families and leads to sparsity in the estimated graphical model. We prove that the maximum likelihood estimator (MLE) in an MTP2 binary exponential family exists if and only if the sign patterns (1,-1) and (-1,1) are represented in the sample for every pair of vertices; in particular, this implies that the MLE may exist with n=d samples, in stark contrast to unrestricted binary exponential families where 2^d samples are required. Finally, we provide a globally convergent algorithm for computing the MLE for MTP2 Ising models similar to iterative proportional scaling and apply it to the analysis of data from two psychological disorders. Throughout, we compare our results on MTP2 Ising models with the Gaussian case and identify similarities and differences.
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