article published yes Shaun Fallat author Steffen Lauritzen author Kayvan Sadeghi author Caroline Uhler author 49ADD78E-F248-11E8-B48F-1D18A9856A870000-0002-7008-0216 Nanny Wermuth author Piotr Zwiernik author CaUh department Gaussian Graphical Models: Theory and Applications project We discuss properties of distributions that are multivariate totally positive of order two (MTP2) related to conditional independence. In particular, we show that any independence model generated by an MTP2 distribution is a compositional semigraphoid which is upward-stable and singleton-transitive. In addition, we prove that any MTP2 distribution satisfying an appropriate support condition is faithful to its concentration graph. Finally, we analyze factorization properties of MTP2 distributions and discuss ways of constructing MTP2 distributions; in particular we give conditions on the log-linear parameters of a discrete distribution which ensure MTP2 and characterize conditional Gaussian distributions which satisfy MTP2. Institute of Mathematical Statistics2017 eng 0090-5364 1510.01290 00040439590000810.1214/16-AOS1478 4531152 - 1184 S. Fallat, S. Lauritzen, K. Sadeghi, C. Uhler, N. Wermuth, and P. Zwiernik, “Total positivity in Markov structures,” <i>Annals of Statistics</i>, vol. 45, no. 3. Institute of Mathematical Statistics, pp. 1152–1184, 2017. Fallat, S., Lauritzen, S., Sadeghi, K., Uhler, C., Wermuth, N., &#38; Zwiernik, P. (2017). Total positivity in Markov structures. <i>Annals of Statistics</i>. Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/16-AOS1478">https://doi.org/10.1214/16-AOS1478</a> Fallat, Shaun, et al. “Total Positivity in Markov Structures.” <i>Annals of Statistics</i>, vol. 45, no. 3, Institute of Mathematical Statistics, 2017, pp. 1152–84, doi:<a href="https://doi.org/10.1214/16-AOS1478">10.1214/16-AOS1478</a>. Fallat S, Lauritzen S, Sadeghi K, Uhler C, Wermuth N, Zwiernik P. Total positivity in Markov structures. <i>Annals of Statistics</i>. 2017;45(3):1152-1184. doi:<a href="https://doi.org/10.1214/16-AOS1478">10.1214/16-AOS1478</a> Fallat, Shaun, Steffen Lauritzen, Kayvan Sadeghi, Caroline Uhler, Nanny Wermuth, and Piotr Zwiernik. “Total Positivity in Markov Structures.” <i>Annals of Statistics</i>. Institute of Mathematical Statistics, 2017. <a href="https://doi.org/10.1214/16-AOS1478">https://doi.org/10.1214/16-AOS1478</a>. Fallat S, Lauritzen S, Sadeghi K, Uhler C, Wermuth N, Zwiernik P. 2017. Total positivity in Markov structures. Annals of Statistics. 45(3), 1152–1184. S. Fallat, S. Lauritzen, K. Sadeghi, C. Uhler, N. Wermuth, P. Zwiernik, Annals of Statistics 45 (2017) 1152–1184. 10892018-12-11T11:50:05Z2025-06-04T08:35:37Z