## 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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