# Total positivity in exponential families with application to binary variables

## TL;DR

This paper explores multivariate totally positive exponential families, highlighting their properties, conditions for maximum likelihood estimation, and applications to graphical models and binary data analysis.

## Contribution

It characterizes MTP2 exponential families, establishes conditions for MLE existence, and introduces a new algorithm for MTP2 Ising models.

## Key findings

- MLE exists with as few as d observations in binary MTP2 models
- MTP2 serves as an implicit regularizer leading to sparsity in graphical models
- Proposed a globally convergent algorithm for MTP2 Ising models

## Abstract

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 $2^d$ 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.