Open Access
June 2017 Total positivity in Markov structures
Shaun Fallat, Steffen Lauritzen, Kayvan Sadeghi, Caroline Uhler, Nanny Wermuth, Piotr Zwiernik
Ann. Statist. 45(3): 1152-1184 (June 2017). DOI: 10.1214/16-AOS1478


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 $\mathrm{MTP}_{2}$.


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Shaun Fallat. Steffen Lauritzen. Kayvan Sadeghi. Caroline Uhler. Nanny Wermuth. Piotr Zwiernik. "Total positivity in Markov structures." Ann. Statist. 45 (3) 1152 - 1184, June 2017.


Received: 1 October 2015; Revised: 1 May 2016; Published: June 2017
First available in Project Euclid: 13 June 2017

zbMATH: 06756077
MathSciNet: MR3662451
Digital Object Identifier: 10.1214/16-AOS1478

Primary: 60E15 , 62H99
Secondary: 15B48

Keywords: association , concentration graph , conditional Gaussian distribution , faithfulness , graphical models , log-linear interactions , Markov property , Positive dependence

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 3 • June 2017
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