Total positivity in Markov structures

By Shaun M. Fallat, Steffen L. Lauritzen, Kayvan Sadeghi, Caroline Uhler, Nanny Wermuth, Piotr Zwiernik. 2015

80 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Computational Theory and Mathematics, Signal Processing

View Paper: https://www.semanticscholar.org/paper/665c2732fd3b917371765f3e062759e613927496

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. B. Schoelkopf
2. Dominik Janzing
3. Caroline Uhler
4. Raj Agrawal
5. Bernhard Schölkopf
6. Jan Peters
7. Joris M. Mooij
8. Kun Zhang

In Computational Theory and Mathematics

1. B. Schoelkopf
2. Raj Agrawal
3. Dominik Janzing
4. Caroline Uhler
5. Uma Roy
6. Bernhard Schölkopf
7. Kun Zhang
8. Jan Peters

In Signal Processing

1. Raj Agrawal
2. Dominik Janzing
3. B. Schoelkopf
4. Caroline Uhler
5. Uma Roy
6. Bernhard Schölkopf
7. David Lopez-Paz

Citations in the corpus

• Detecting Confounding in Multivariate Linear Models via Spectral Analysis (2017, background): “8 Note also the concept of multivariate total positivity of order two (MTP2) [34], which implies positive correlations between all variables and occurs in many applications [35] such as Markov random fields that consist only of attractive interaction terms.”
• Covariance Matrix Estimation under Total Positivity for Portfolio Selection* (2019, background): “MTP 2 was first studied in [FKG71, KR80a, Bøl82, KR83] from a purely theoretical perspective and later also in the context of statistical modeling, in particular graphical models, in [SH14, FLS + 17, LUZ19a, LUZ19b].”