Max-linear models on directed acyclic graphs

TL;DR

This paper introduces a max-linear recursive structural equation model on directed acyclic graphs, characterizing its coefficients, structure, and minimal graph representation, advancing understanding of max-linear causal models.

Contribution

It provides a detailed characterization of max-linear models generated by recursive structural equations, including a fixed point solution for the coefficient matrix and a minimal DAG representation.

Findings

Max-linear coefficient matrix is the solution of a fixed point equation.

A unique minimal DAG represents the recursive structural equations.

The model introduces a natural order among variables and coefficients.

Abstract

We consider a new recursive structural equation model where all variables can be written as max-linear function of their parental node variables and independent noise variables. The model is max-linear in terms of the noise variables, and its causal structure is represented by a directed acyclic graph. We detail the relation between the weights of the recursive structural equation model and the coefficients in its max-linear representation. In particular, we characterize all max-linear models which are generated by a recursive structural equation model, and show that its max-linear coefficient matrix is the solution of a fixed point equation. We also find a unique minimum directed acyclic graph representing the recursive structural equations of the variables. The model structure introduces a natural order between the node variables and the max-linear coefficients. This yields representations of the vector components, which are based on a minimum number of node and noise variables.