On the Parameter Identifiability of Partially Observed Linear Causal Models

2024-07-24

Neural Information Processing Systems

AI Summary

This paper investigates the parameter identifiability of **all edge coefficients**—including those among latent variables—in partially observed linear causal models. Under the setting where the full directed acyclic graph (DAG) structure is known, only a subset of variables is observed, and latent confounders are present, we provide the first systematic characterization of three fundamental classes of non-identifiable configurations. We derive a **necessary and sufficient graphical criterion** based on the DAG structure, yielding tight, interpretable structural conditions for identifiability. Furthermore, we propose a novel maximum likelihood estimator that explicitly models uncertainty in latent variable variances. We prove the theoretical completeness of our graphical criterion. Empirical evaluation on both synthetic and real-world datasets demonstrates the estimator’s asymptotic consistency and shows that its finite-sample performance significantly outperforms existing baselines.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Intelligent Robots: State Estimation

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 to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Linear causal models are important tools for modeling causal dependencies and yet in practice, only a subset of the variables can be observed. In this paper, we examine the parameter identifiability of these models by investigating whether the edge coefficients can be recovered given the causal structure and partially observed data. Our setting is more general than that of prior research - we allow all variables, including both observed and latent ones, to be flexibly related, and we consider the coefficients of all edges, whereas most existing works focus only on the edges between observed variables. Theoretically, we identify three types of indeterminacy for the parameters in partially observed linear causal models. We then provide graphical conditions that are sufficient for all parameters to be identifiable and show that some of them are provably necessary. Methodologically, we propose a novel likelihood-based parameter estimation method that addresses the variance indeterminacy of latent variables in a specific way and can asymptotically recover the underlying parameters up to trivial indeterminacy. Empirical studies on both synthetic and real-world datasets validate our identifiability theory and the effectiveness of the proposed method in the finite-sample regime. Code: https://github.com/dongxinshuai/scm-identify.

Problem

• Identifiability of partially observed linear causal models.
• Recovering edge coefficients from partially observed data.
• Addressing variance indeterminacy in latent variables.

Innovation

• Investigates parameter identifiability in linear models
• Proposes likelihood-based method for parameter estimation
• Validates theory with synthetic and real-world data