Maximum likelihood estimation in Gaussian models under total positivity

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  • AOS1668

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Steffen L. Lauritzen, Caroline Uhler, Piotr Zwiernik

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.
OriginalsprogEngelsk
TidsskriftAnnals of Statistics
Vol/bind47
Udgave nummer4
Sider (fra-til)1835-1863
ISSN0090-5364
DOI
StatusUdgivet - 21 maj 2019

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