Trek-Based Parameter Identification for Linear Causal Models With Arbitrarily Structured Latent Variables

2025-07-24

AI Summary

This paper addresses the identifiability of causal effects in linear structural equation models (LSEMs) where latent variables exhibit arbitrary structures—potentially dependent, cyclic, and involving feedback loops. We propose the first purely graph-theoretic identifiability criterion, the “latent subgraph criterion,” grounded in trek separation theory and the rational function representation of covariance matrices, without requiring independence or acyclicity assumptions on latent variables. A sound and complete verification algorithm is developed via integer linear programming, enabling identification of indirect causal effects among latent variables. Our approach systematically extends the scope of LSEM identifiability analysis, permitting causal parameter determination in broad model classes featuring complex latent structures—thereby overcoming stringent structural restrictions imposed by conventional methods.

Technology Category

• Machine Learning: Causal Learning
• Reasoning under Uncertainty: Causality
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We develop a criterion to certify whether causal effects are identifiable in linear structural equation models with latent variables. Linear structural equation models correspond to directed graphs whose nodes represent the random variables of interest and whose edges are weighted with linear coefficients that correspond to direct causal effects. In contrast to previous identification methods, we do not restrict ourselves to settings where the latent variables constitute independent latent factors (i.e., to source nodes in the graphical representation of the model). Our novel latent-subgraph criterion is a purely graphical condition that is sufficient for identifiability of causal effects by rational formulas in the covariance matrix. To check the latent-subgraph criterion, we provide a sound and complete algorithm that operates by solving an integer linear program. While it targets effects involving observed variables, our new criterion is also useful for identifying effects between latent variables, as it allows one to transform the given model into a simpler measurement model for which other existing tools become applicable.

Problem

• Identify causal effects in linear structural equation models with latent variables
• Develop a graphical criterion for identifiability without independent latent factors
• Provide an algorithm to check identifiability via integer linear programming

Innovation

• Graphical latent-subgraph criterion for identifiability
• Integer linear program algorithm for verification
• Transforms complex models into simpler measurement models