We study exponential families of distributions that are multivariate totally
positive of order 2 (MTP2), show that these are convex exponential families
and derive conditions for existence of the MLE. Quadratic exponential
familes of MTP2 distributions contain attractive Gaussian graphical models
and ferromagnetic Ising models as special examples. We show that these 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 both of the sign patterns (1,−1) and (−1, 1) are represented
in the sample for every pair of variables; in particular, this implies that
the MLE may exist with n = d observations, in stark contrast to unrestricted
binary exponential families where 2d observations are required. Finally, we
provide a novel and 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.