Distributional Equivalence in Linear Non-Gaussian Latent-Variable Cyclic Causal Models: Characterization and Learning

TL;DR

This paper characterizes when two linear non-Gaussian latent-variable cyclic causal models are distributionally equivalent, enabling structure-free causal discovery through a novel graphical criterion and an algorithm to identify models from data.

Contribution

It introduces the first equivalence characterization for latent-variable models without structural assumptions, advancing causal discovery methods.

Findings

Established graphical criterion for distributional equivalence.

Developed edge rank constraints as a new tool.

Created an algorithm to recover models up to equivalence.

Abstract

Causal discovery with latent variables is a fundamental task. Yet most existing methods rely on strong structural assumptions, such as enforcing specific indicator patterns for latents or restricting how they can interact with others. We argue that a core obstacle to a general, structural-assumption-free approach is the lack of an equivalence characterization: without knowing what can be identified, one generally cannot design methods for how to identify it. In this work, we aim to close this gap for linear non-Gaussian models. We establish the graphical criterion for when two graphs with arbitrary latent structure and cycles are distributionally equivalent, that is, they induce the same observed distribution set. Key to our approach is a new tool, edge rank constraints, which fills a missing piece in the toolbox for latent-variable causal discovery in even broader settings. We further provide a procedure to traverse the whole equivalence class and develop an algorithm to recover models from data up to such equivalence. To our knowledge, this is the first equivalence characterization with latent variables in any parametric setting without structural assumptions, and hence the first structural-assumption-free discovery method. Code and an interactive demo are available at https://equiv.cc.