Identifiability and estimation of recursive max-linear models

Gissibl, Nadine; Klüppelberg, Claudia; Lauritzen, Steffen

doi:10.48550/arxiv.1901.03556

Cited by 3 publications

(10 citation statements)

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“…We have so far in this article not discussed identifiability, estimation, or any other statistical issues associated with these models. These have been briefly considered in [17]; see also [15]. This work was extended to a recursive max-linear model with propagating noise in [7], but we are not considering models with noise in this article.…”

Section: Discussionmentioning

confidence: 99%

Conditional Independence in Max-linear Bayesian Networks

Améndola,

Klüppelberg,

Lauritzen

et al. 2020

Preprint

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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.

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Section: Discussionmentioning

confidence: 99%

Conditional Independence in Max-linear Bayesian Networks

Améndola,

Klüppelberg,

Lauritzen

et al. 2020

Preprint

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“…Misra and Kuruoglu [20] consider stable noise variables in a Bayesian network and develop a structure learning algorithm based on BIC. The work of Gissibl et al [11] also studies causal questions related to extreme events. They consider max-linear models [10] where only the largest effect propagates to the descendants in a Bayesian network.…”

Section: Introduction and Background 1introductionmentioning

confidence: 99%

Causal discovery in heavy-tailed models

Gnecco¹,

Meinshausen²,

Peters³

et al. 2019

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Causal questions are omnipresent in many scientific problems. While much progress has been made in the analysis of causal relationships between random variables, these methods are not well suited if the causal mechanisms manifest themselves only in extremes. This work aims to connect the two fields of causal inference and extreme value theory. We define the causal tail coefficient that captures asymmetries in the extremal dependence of two random variables. In the population case, the causal tail coefficient is shown to reveal the causal structure if the distribution follows a linear structural causal model. This holds even in the presence of latent common causes that have the same tail index as the observed variables. Based on a consistent estimator of the causal tail coefficient, we propose a computationally highly efficient algorithm that infers causal structure from finitely many data. We prove that our method consistently estimates the causal order and compare it to other well-established and non-extremal approaches in causal discovery on synthetic data. The code is available as an open-access R package on Github.

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“…. , X d ) (see [14], Theorem 1). Also, B is idempotent with respect to the tropical matrix multiplication defined in (2.4) below, and defines a graphical model on a DAG.…”

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confidence: 97%

“…However, while the ML coefficient matrix B of X is identifiable from the distribution of X, the edge weight matrix C is generally not, see Theorem 5.4(b) in [13]. Theorem 5.3 in that paper and Theorem 2 in [14] show that an edge with edge weight c ji is identifiable from B if and only if it is the unique path from j to i with d ji (p) = b ji .…”

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confidence: 98%

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Recursive max-linear models with propagating noise

Buck¹,

Klüppelberg²

2020

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Recursive max-linear vectors model causal dependence between node variables by a structural equation model, expressing each node variable as a max-linear function of its parental nodes in a directed acyclic graph (DAG) and some exogenous innovation. For such a model, there exists a unique minimum DAG, represented by the Kleene star matrix of its edge weight matrix, which identifies the model and can be estimated. For a more realistic statistical modeling we introduce some random observational noise. A probabilistic analysis of this new noisy model reveals that the unique minimum DAG representing the distribution of the non-noisy model remains unchanged and identifiable. Moreover, the distribution of the minimum ratio estimators of the model parameters at their left limits are completely determined by the distribution of the noise variables up to a positive constant. Under a regular variation condition on the noise variables we prove that the estimated Kleene star matrix converges to a matrix of independent Weibull entries after proper centering and scaling.

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Identifiability and estimation of recursive max-linear models

Cited by 3 publications

References 0 publications

Conditional Independence in Max-linear Bayesian Networks

Conditional Independence in Max-linear Bayesian Networks

Causal discovery in heavy-tailed models

Recursive max-linear models with propagating noise

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