Maximum likelihood estimation in Gaussian models under total positivity
Name
1702.04031.pdf
Description
Accepted version
Size
985.94 KB
Format
Unknown
Checksum (MD5)
871577dff1ddca4dbda1f41d7061177e
Author(s)
Lauritzen, Steffen
Uhler, Caroline
Zweirnik, Piotr
Date Issued
2019
Journal
The Annals of Statistics
Publisher
Institute of Mathematical Statistics
Version
Author's final manuscript
Abstract
© 2019 Institute of Mathematical Statistics. 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.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1214/17-AOS1668
