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 $\mathrm{MTP}_{2}$.
Citation
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. https://doi.org/10.1214/16-AOS1478
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