Identifiability in Graphical Discrete Lyapunov Models

TL;DR

This paper investigates the identifiability of parameters in discrete Lyapunov models with non-Gaussian errors, establishing conditions for unique parameter recovery and exploring model equivalence for certain graph structures.

Contribution

It introduces new identifiability results for discrete Lyapunov models with non-Gaussian errors, including rational expressions of parameters and initial insights into model equations.

Findings

Generic identifiability for directed acyclic graphs with self-loops

Parameters expressed as rational functions of cumulants

Model equivalence results for specific graph classes

Abstract

In this paper, we study discrete Lyapunov models, which consist of steady-state distributions of first-order vector autoregressive models. The parameter matrix of such a model encodes a directed graph whose vertices correspond to the components of the random vector. This combinatorial framework naturally allows for cycles in the graph structure. We focus on the fundamental problem of identifying the entries of the parameter matrix. In contrast to the classical setting, we assume non-Gaussian error terms, which allows us to use the higher-order cumulants of the model. In this setup, we show generic identifiability for directed acyclic graphs with self-loops at each vertex and show how to express the parameters as a rational function of the cumulants. Furthermore, we establish local identifiability for all directed graphs containing self loops at each vertex and no isolated vertices. Finally, we provide first results on the defining equations of the models, showing model equivalence for certain graphs and paving the way towards structure learning.