Total positivity in Markov structures
Shaun Fallat, Steffen Lauritzen, Kayvan Sadeghi, Caroline Uhler, Nanny Wermuth, Piotr Zwiernik
The Annals of Statistics · 2017 · Volume 45 · Issue 3
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 $\mathrm{MTP}_{2}$.
Keywords
15B48, 60E15, 62H99, association, concentration graph, conditional Gaussian distribution, faithfulness, graphical models, log-linear interactions, Markov property, Positive dependence
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