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Total positivity in Markov structures. (English) Zbl 1414.60010

Summary: 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 \(\mathrm{MTP}_{2}\).

MSC:

60E15 Inequalities; stochastic orderings
62H99 Multivariate analysis
15B48 Positive matrices and their generalizations; cones of matrices
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