Total positivity in Markov structures

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

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Total positivity in Markov structures

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arxiv 1510.01290 v2 pith:ZHKM3BRY submitted 2015-10-05 math.ST stat.TH

Shaun Fallat , Steffen Lauritzen , Kayvan Sadeghi , Caroline Uhler , Nanny Wermuth , Piotr Zwiernik This is my paper

classification math.STstat.TH

keywords mtp2distributionsdistributionconditionaldiscussindependenceparticularproperties

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We discuss properties of distributions that are multivariate totally positive of order two (MTP2) related to conditional independence. In particular, we show that any independence model generated by an MTP2 distribution is a compositional semigraphoid which is upward-stable and singleton-transitive. In addition, we prove that any MTP2 distribution satisfying an appropriate support condition is faithful to its concentration graph. Finally, we analyze factorization properties of MTP2 distributions and discuss ways of constructing MTP2 distributions; in particular we give conditions on the log-linear parameters of a discrete distribution which ensure MTP2 and characterize conditional Gaussian distributions which satisfy MTP2.

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