We discuss properties of distributions that are multivariate totally positive of order two (MTP 2 ) related to conditional independence. In particular, we show that any independence model generated by an MTP 2 distribution is a compositional semi-graphoid which is upward-stable and singletontransitive. In addition, we prove that any MTP 2 distribution satisfying an appropriate support condition is faithful to its concentration graph. Finally, we analyze factorization properties of MTP 2

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... tions and discuss ways of constructing MTP 2 distributions; in particular, we give conditions on the log-linear parameters of a discrete distribution which ensure MTP 2 and characterize conditional Gaussian distributions which satisfy MTP 2 .