Your dreams may come true with MTP2

We study maximum likelihood estimation for exponential families that are multivariate totally positive of order two (MTP2). Such distributions appear in the context of ferromagnetism in the Ising model and various latent models, as for example Brownian motion tree models used in phylogenetics. We show that maximum likelihood estimation for MTP2 exponential families is a convex optimization problem. For quadratic exponential families such as Ising models and Gaussian graphical models, we show that MTP2 implies sparsity of the underlying graph without the need of a tuning parameter. In addition, we characterize a subgraph and a supergraph of Gaussian graphical models under MTP2. Moreover, we show that the MLE always exists even in the high-dimensional setting. These properties make MTP2 constraints an intriguing alternative to methods for learning sparse graphical models such as the graphical lasso.