# Square Root Graphical Models: Multivariate Generalizations of Univariate Exponential Families that Permit Positive Dependencies

#### David Inouye, Pradeep Ravikumar, Inderjit Dhillon

Abstract:   We develop a novel class of parametric graphical models, called Square Root Graphical Models (SQR), that provides multivariate generalizations of univariate exponential family distributions—e.g. discrete, Gaussian, exponential and Poisson distributions. Previous multivariate graphical models [Yang et al. 2015] did not allow positive dependencies for the exponential and Poisson generalizations. However, in many real-world datasets, variables clearly have positive dependencies. For example, the airport delay time in New York—modeled as an exponential distribution—is positively related to the delay time in Boston. With this motivation, we give an example of our model class derived from the univariate exponential distribution that allows for almost arbitrary positive and negative dependencies with only a mild condition on the parameter matrix—a condition akin to the positive definiteness of the Gaussian covariance matrix. Our Poisson generalization allows for both positive and negative dependencies without any constraints on the parameter values. We also develop parameter estimation methods using node-wise regressions with $ell_1$ regularization and likelihood approximation methods using sampling. Finally, we demonstrate our exponential generalization on a dataset of airport delay times.