Markov equivalence of max-linear Bayesian networks

Max-linear Bayesian networks have emerged as highly applicable models for causal inference via extreme value data. However, conditional independence (CI) for max-linear Bayesian networks behaves differently than for classical Gaussian Bayesian networks. We establish the parallel between the two theories via tropicalization, and establish the surprising result that the Markov equivalence classes for max-linear Bayesian networks coincide with the ones obtained by regular CI. Our paper opens up many problems at the intersection of extreme value statistics, causal inference and tropical geometry.

[1]  Dan Geiger,et al.  Identifying independence in bayesian networks , 1990, Networks.

[2]  Yaroslav Shitov.  The complexity of tropical matrix factorization , 2012 .

[3]  Ngoc Mai Tran.  Polytropes and Tropical Eigenspaces: Cones of Linearity , 2014, Discret. Comput. Geom..

[4]  Pauli Miettinen,et al.  Algorithms for approximate subtropical matrix factorization , 2017, Data Mining and Knowledge Discovery.

[5]  Michael I. Jordan.  Graphical Models , 2003 .

[6]  Judea Pearl,et al.  An Algorithm for Deciding if a Set of Observed Independencies Has a Causal Explanation , 1992, UAI.

[7]  Claudia Kluppelberg,et al.  Recursive max-linear models with propagating noise , 2020, Electronic Journal of Statistics.

[8]  Caroline Uhler,et al.  Counting Markov equivalence classes for DAG models on trees , 2017, Discret. Appl. Math..

[9]  S. Lauritzen,et al.  Bayesian Networks for Max-Linear Models , 2019, Network Science.

[10]  Claudia Klüppelberg,et al.  Estimating an extreme Bayesian network via scalings , 2019, J. Multivar. Anal..

[11]  Caroline Uhler,et al.  Counting Markov Equivalence Classes by Number of Immoralities , 2016, UAI.

[12]  S. Sullivant,et al.  Trek separation for Gaussian graphical models , 2008, 0812.1938.

[13]  Claudia Klüppelberg,et al.  Causal Discovery of a River Network from its Extremes , 2021, ArXiv.

[14]  Tom Burr,et al.  Causation, Prediction, and Search , 2003, Technometrics.

[15]  Christopher Meek,et al.  Strong completeness and faithfulness in Bayesian networks , 1995, UAI.

[16]  Seth Sullivant,et al.  Algebraic statistics , 2018, ISSAC.

[17]  S. Lauritzen,et al.  Identifiability and estimation of recursive max‐linear models , 2019, Scandinavian Journal of Statistics.

[18]  Phyllis Wan,et al.  $k$-means clustering of extremes , 2019, Electronic Journal of Statistics.

[19]  Masaaki Sibuya,et al.  Bivariate extreme statistics, I , 1960 .

[20]  Judea Pearl,et al.  A Theory of Inferred Causation , 1991, KR.

[21]  Anthony C. Davison,et al.  Extremes on river networks , 2015, 1501.02663.

[22]  Nadine Gissibl.  Graphical modeling of extremes , 2018 .

[23]  B. Sturmfels,et al.  ELIMINATION THEORY FOR TROPICAL VARIETIES , 2007, 0704.3471.

[24]  C. Kluppelberg,et al.  Max-linear models on directed acyclic graphs , 2015, Bernoulli.

[25]  Dan Geiger,et al.  d-Separation: From Theorems to Algorithms , 2013, UAI.

[26]  D. Madigan,et al.  A characterization of Markov equivalence classes for acyclic digraphs , 1997 .

[27]  Michael D. Perlman,et al.  Enumerating Markov Equivalence Classes of Acyclic Digraph Models , 2001, UAI.

[28]  Judea Pearl,et al.  Equivalence and Synthesis of Causal Models , 1990, UAI.

[29]  Sebastian Engelke,et al.  Graphical models for extremes , 2018, Journal of the Royal Statistical Society: Series B (Statistical Methodology).