Conditional Independence in Max-linear Bayesian Networks

Abstract

Motivated by extreme value theory, max-linear Bayesian networks have been recently introduced and studied as an alternative to linear structural equation models. However, for max-linear systems the classical independence results for Bayesian networks are far from exhausting valid conditional independence statements. We use tropical linear algebra to derive a compact representation of the conditional distribution given a partial observation, and exploit this to obtain a complete description of all conditional independence relations. In the context-dependent case, where conditional independence is queried relative to a specific value of the conditioning variables, we introduce the notion of a source DAG to disclose the valid conditional independence relations. In the context-independent case we characterize conditional independence through a modified separation concept, * -separation, combined with a tropical eigenvalue condition. We also introduce the notion of an impact graph which describes how extreme events spread deterministically through the network and we give a complete characterization of such impact graphs. Our analysis opens up several interesting questions concerning conditional independence and tropical geometry.

Discrete Max-Linear Bayesian Networks

“…Graphs, Posets, and Lattices. Our notation for graphs follows that of [1] while our notation for posets and lattices follows that of [11]. The graphs in this note will primarily be directed graphs which are given by a vertex set V = {1, .…”

“…We refer the reader to [12, Chapter 13] for additional information on parameterizations of graphical models. We first form a DAG with edges i → j if i < j is a cover relation in P and associate a binary random variable X i to each i ∈ P. Then the CBN model is a directed graphical model on the X i with conditional probabilities given by (1) (P(…”

“…Let ψ be the map parameterizing the 2-state D-MLBN model on D. Then image(ρ) is equal to image(ψ) after a natural relabeling of coordinates. In particular, there exists a bijection, φ : J(D tr ) → G(D, 2) such that ρ g (θ (1) 0 , θ (1) 1 , . .…”

“…where ∨ denotes maximum and pa(i) denotes the parents of vertex i in D. These models are able to more accurately model extreme events spreading throughout a network than standard discrete or Gaussian Bayesian networks [6,10] and also have interesting ties to tropical geometry [1]. Max-linear models also exhibit a remarkable property concerning conditional independence: they are usually not faithful to their underlying DAG, meaning that they often satisfy more conditional independence statements than those implied by the d-separation criterion [10].…”

“…Max-linear models also exhibit a remarkable property concerning conditional independence: they are usually not faithful to their underlying DAG, meaning that they often satisfy more conditional independence statements than those implied by the d-separation criterion [10]. Améndola, Klüppelberg, Lauritzen, and Tran recently gave a new criterion named * -separation which gives a complete set of conditional independence statements for maxlinear models [1]. There has also been work done to establish when the parameters of the model are identifiable [6,7,10].…”