The Annals of Statistics

Maximum likelihood estimation in Gaussian models under total positivity

Steffen Lauritzen, Caroline Uhler, and Piotr Zwiernik

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Abstract

We analyze the problem of maximum likelihood estimation for Gaussian distributions that are multivariate totally positive of order two ($\mathrm{MTP}_{2}$). 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 $n\geq2$ 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 $\mathrm{MTP}_{2}$ 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 $\mathrm{MTP}_{2}$ constraint which are structurally similar to iterative proportional scaling. We conclude the paper with a discussion of signed $\mathrm{MTP}_{2}$ distributions.

Article information

Source
Ann. Statist., Volume 47, Number 4 (2019), 1835-1863.

Dates
Received: February 2017
Revised: November 2017
First available in Project Euclid: 21 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aos/1558425632

Digital Object Identifier
doi:10.1214/17-AOS1668

Mathematical Reviews number (MathSciNet)
MR3953437

Zentralblatt MATH identifier
07082272

Subjects
Primary: 60E15: Inequalities; stochastic orderings 62H99: None of the above, but in this section
Secondary: 15B48: Positive matrices and their generalizations; cones of matrices

Keywords
$\mathrm{MTP}_{2}$ distribution attractive Gaussian Markov random field (GMRF) nonfrustrated GRMF Gaussian graphical model inverse M-matrix ultrametric

Citation

Lauritzen, Steffen; Uhler, Caroline; Zwiernik, Piotr. Maximum likelihood estimation in Gaussian models under total positivity. Ann. Statist. 47 (2019), no. 4, 1835--1863. doi:10.1214/17-AOS1668. https://projecteuclid.org/euclid.aos/1558425632


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