The Annals of Statistics

Total positivity in Markov structures

Shaun Fallat, Steffen Lauritzen, Kayvan Sadeghi, Caroline Uhler, Nanny Wermuth, and Piotr Zwiernik

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Abstract

We discuss properties of distributions that are multivariate totally positive of order two ($\mathrm{MTP}_{2}$) related to conditional independence. In particular, we show that any independence model generated by an $\mathrm{MTP}_{2}$ distribution is a compositional semi-graphoid which is upward-stable and singleton-transitive. In addition, we prove that any $\mathrm{MTP}_{2}$ distribution satisfying an appropriate support condition is faithful to its concentration graph. Finally, we analyze factorization properties of $\mathrm{MTP}_{2}$ distributions and discuss ways of constructing $\mathrm{MTP}_{2}$ distributions; in particular, we give conditions on the log-linear parameters of a discrete distribution which ensure $\mathrm{MTP}_{2}$ and characterize conditional Gaussian distributions which satisfy $\mathrm{MTP}_{2}$.

Article information

Source
Ann. Statist., Volume 45, Number 3 (2017), 1152-1184.

Dates
Received: October 2015
Revised: May 2016
First available in Project Euclid: 13 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.aos/1497319691

Digital Object Identifier
doi:10.1214/16-AOS1478

Mathematical Reviews number (MathSciNet)
MR3662451

Zentralblatt MATH identifier
06756077

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
Association concentration graph conditional Gaussian distribution faithfulness graphical models log-linear interactions Markov property positive dependence

Citation

Fallat, Shaun; Lauritzen, Steffen; Sadeghi, Kayvan; Uhler, Caroline; Wermuth, Nanny; Zwiernik, Piotr. Total positivity in Markov structures. Ann. Statist. 45 (2017), no. 3, 1152--1184. doi:10.1214/16-AOS1478. https://projecteuclid.org/euclid.aos/1497319691


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