Total Positivity in Markov Structures

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 semigraphoid which is upward-stable and singleton-transitive. 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$_{2}$ distributions 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$_{2}$.SF was supported in part by an NSERC Discovery Research Grant, KS by grant #FA9550-12-1-0392 from the U.S. Air Force Office of Scientic Research (AFOSR) and the Defense Advanced Research Projects Agency (DARPA), CU by the Austrian Science Fund (FWF) Y 903-N35, and PZ by the European Union Seventh Framework Programme PIOF-GA-2011-300975