Journal Article

MAXIMUM LIKELIHOOD ESTIMATION IN GAUSSIAN MODELS UNDER TOTAL POSITIVITY

Lauritzen, Steffen, Uhler, Caroline, Zwiernik, Piotr

2019

Overview

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 n ≥ 2 observations, irrespective of the underlying dimension. Slawski and Hein [Linear Algebra Appl. 473 (2015) 145–179], 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.

• Publisher: Institute of Mathematical Statistics
• Subject: Algorithms / Asymptotic methods / Clustering / Correlation analysis / Covariance matrix / Empirical analysis / Estimating techniques / Graph theory / Iterative methods / Linear algebra / Maximum likelihood estimation / Maximum likelihood estimators / Maximum likelihood method / Normal distribution / Studies / Upper bounds