Total positivity in Markov structures

TL;DR

This paper explores the properties of multivariate totally positive of order two (MTP2) distributions, their independence models, and their factorization, providing new insights into their structure and construction methods.

Contribution

It establishes that MTP2 distributions generate compositional semigraphoids, are upward-stable and singleton-transitive, and characterizes conditions for their faithfulness and construction.

Findings

MTP2 distributions form compositional semigraphoids.

MTP2 distributions are upward-stable and singleton-transitive.

Conditions for MTP2 in discrete and Gaussian distributions.

Abstract

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.