Open Access
August 2019 Maximum likelihood estimation in Gaussian models under total positivity
Steffen Lauritzen, Caroline Uhler, Piotr Zwiernik
Ann. Statist. 47(4): 1835-1863 (August 2019). DOI: 10.1214/17-AOS1668


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.


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Steffen Lauritzen. Caroline Uhler. Piotr Zwiernik. "Maximum likelihood estimation in Gaussian models under total positivity." Ann. Statist. 47 (4) 1835 - 1863, August 2019.


Received: 1 February 2017; Revised: 1 November 2017; Published: August 2019
First available in Project Euclid: 21 May 2019

zbMATH: 07082272
MathSciNet: MR3953437
Digital Object Identifier: 10.1214/17-AOS1668

Primary: 60E15 , 62H99
Secondary: 15B48

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

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 4 • August 2019
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