# Maximum likelihood estimation in Gaussian models under total positivity

## TL;DR

This paper studies maximum likelihood estimation for Gaussian models with total positivity constraints, proving existence with minimal data, and introduces algorithms for efficient estimation and analysis of the resulting sparse inverse covariance structures.

## Contribution

It provides a simple proof of MLE existence for MTP2 Gaussian models with minimal observations and develops globally convergent algorithms for estimation under these constraints.

## Key findings

- MLE exists with at least 2 observations regardless of dimension
- The ML graph can be bounded using maximum weight spanning forest
- Algorithms similar to iterative proportional scaling are proposed

## Abstract

We analyze the problem of maximum likelihood estimation for Gaussian distributions that are multivariate totally positive of order two (MTP2). By exploiting connections to phylogenetics and single-linkage clustering, we give a simple proof that the maximum likelihood estimator (MLE) for such distributions exists based on at least 2 observations, irrespective of the underlying dimension. Slawski and Hein, who first proved this result, also provided empirical evidence showing that the MTP2 constraint serves as an implicit regularizer and leads to sparsity in the estimated inverse covariance matrix, determining what we name the ML graph. We show that we can find an upper bound for the ML graph by adding edges corresponding to correlations in excess of those explained by the maximum weight spanning forest of the correlation matrix. Moreover, we provide globally convergent coordinate descent algorithms for calculating the MLE under the MTP2 constraint which are structurally similar to iterative proportional scaling. We conclude the paper with a discussion of signed MTP2 distributions.