TY - JOUR

T1 - Total positivity in exponential families with application to binary variables

AU - Lauritzen, Steffen L.

AU - Uhler, Caroline

AU - Zwiernik, Piotr

PY - 2021

Y1 - 2021

N2 - We study exponential families of distributions that are multivariate totallypositive of order 2 (MTP2), show that these are convex exponential familiesand derive conditions for existence of the MLE. Quadratic exponentialfamiles of MTP2 distributions contain attractive Gaussian graphical modelsand ferromagnetic Ising models as special examples. We show that these aredefined by intersecting the space of canonical parameters with a polyhedralcone whose faces correspond to conditional independence relations. HenceMTP2 serves as an implicit regularizer for quadratic exponential familiesand leads to sparsity in the estimated graphical model. We prove that themaximum likelihood estimator (MLE) in an MTP2 binary exponential familyexists if and only if both of the sign patterns (1,−1) and (−1, 1) are representedin the sample for every pair of variables; in particular, this implies thatthe MLE may exist with n = d observations, in stark contrast to unrestrictedbinary exponential families where 2d observations are required. Finally, weprovide a novel and globally convergent algorithm for computing the MLEfor MTP2 Ising models similar to iterative proportional scaling and apply itto the analysis of data from two psychological disorders.

AB - We study exponential families of distributions that are multivariate totallypositive of order 2 (MTP2), show that these are convex exponential familiesand derive conditions for existence of the MLE. Quadratic exponentialfamiles of MTP2 distributions contain attractive Gaussian graphical modelsand ferromagnetic Ising models as special examples. We show that these aredefined by intersecting the space of canonical parameters with a polyhedralcone whose faces correspond to conditional independence relations. HenceMTP2 serves as an implicit regularizer for quadratic exponential familiesand leads to sparsity in the estimated graphical model. We prove that themaximum likelihood estimator (MLE) in an MTP2 binary exponential familyexists if and only if both of the sign patterns (1,−1) and (−1, 1) are representedin the sample for every pair of variables; in particular, this implies thatthe MLE may exist with n = d observations, in stark contrast to unrestrictedbinary exponential families where 2d observations are required. Finally, weprovide a novel and globally convergent algorithm for computing the MLEfor MTP2 Ising models similar to iterative proportional scaling and apply itto the analysis of data from two psychological disorders.

U2 - 10.1214/20-AOS2007

DO - 10.1214/20-AOS2007

M3 - Journal article

VL - 49

SP - 1436

EP - 1459

JO - Annals of Statistics

JF - Annals of Statistics

SN - 0090-5364

ER -