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Name: Frank Rottger
Talk Title: Total positivity in graphical extremes
Abstract: \documentclass[10pt,a4paper]{article}\usepackage{amsmath}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{graphicx}\usepackage[round]{natbib}\begin{document}\section*{Total positivity in graphical extremes}\cite{EH2020} introduce a general theory for conditional independence and graphical models for extremes. Similarly to Gaussian graphical models, they show that for H\"usler--Reiss graphical models, conditional independencies are encoded through a transformation of the corresponding parameter matrix~$\Gamma$.
Multivariate total positivity of order 2 ($ \text{MTP}_2 $) is a strong form of positive dependence that induces many interesting properties in graphical modeling.
A multivariate Gaussian distribution is $ \text{MTP}_2 $ when its precision matrix is an M-matrix, i.e., when all the non-diagonal entries in the inverse covariance matrix are non-positive.
We show that a H\"usler--Reiss distribution is $\text{MTP}_2 $ if and only if the inverse Farris transform of the parameter matrix $\Gamma$ is the inverse of a diagonally dominant M-matrix.
Furthermore, we prove that an extremal tree model is $ \text{MTP}_2 $ if and only if its bivariate marginals are $ \text{MTP}_2 $.
This implies that all H\"usler--Reiss tree models are $ \text{MTP}_2 $, while in comparison Gaussian tree models are only $ \text{MTP}_2 $ if their covariance matrices are non-negative.
Similar to \cite{LUZ2019} we construct a coordinate descent algorithm to find a pseudo maximum likelihood estimator under the $ \text{MTP}_2$ constraint and show that it enforces a sparse extremal graphical model. We apply this method to simulated and real data.
This is joint work with Sebastian Engelke and Piotr Zwiernik.
\bibliographystyle{plainnat}
\begin{thebibliography}{2}
\bibitem[Engelke and Hitz(2020)]{EH2020}
Sebastian Engelke and Adrien~S. Hitz.
\newblock Graphical models for extremes.
\newblock \emph{Journal of the Royal Statistical Society: Series B (Statistical
Methodology)}, 82\penalty0 (4):\penalty0 871--932, 2020.
\bibitem[Lauritzen et~al.(2019)Lauritzen, Uhler, and Zwiernik]{LUZ2019}
Steffen Lauritzen, Caroline Uhler, and Piotr Zwiernik.
\newblock Maximum likelihood estimation in {G}aussian models under total positivity.
\newblock \emph{Ann. Statist.}, 47\penalty0 (4):\penalty0 1835--1863, 2019.
\end{thebibliography}
\end{document}
This talk is a contributed talk at EVA 2021. View the programme here.