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Name: Stefka Asenova

Talk Title: Extremes of Markov random fields on block graphs

Abstract: We study the joint occurrence of large values of a Markov random field or undirected graphical model associated to a block graph. Such graphs containing trees as special cases, we aim to generalize recent results for extremes of Markov trees. Every pair of nodes in a block graph is connected by a unique shortest path. These paths are shown to determine the limiting distribution of the properly rescaled random field given that a fixed variable exceeds a high threshold.  When the said limit is a Hüsler--Reiss multivariate Pareto distribution, the local Markov property of the original field induces a particular structure on the limiting parameter matrix. Thanks to these algebraic relations, the parameters are still identifiable even if some variables are latent. The precise identifiability criterion turns out to be close to the one for Markov trees.  Still on block graphs, we study an additive factor graphical model with respect to a directed acyclic graph. The model has the same form as a max-linear Bayesian network but with maxima replaced by sums. The factor variables are assumed to be heavy-tailed and absolutely continuous, properties which then transfer to the joint distribution. We give a sufficient condition on the graph structure for such a Bayesian network to also satisfy the global Markov property with respect to the undirected graph. The latter property then guarantees that the joint extremes of the model have the special form induced by the shortest (undirected) paths connecting nodes. The identifiability of the parameters of the limit distribution in case some variables are latent is discussed as well.

This talk is a contributed talk at EVA 2021.