Causal Discovery in Linear Models with Unobserved Variables and Measurement Error (2407.19426v1)

Abstract: The presence of unobserved common causes and the presence of measurement error are two of the most limiting challenges in the task of causal structure learning. Ignoring either of the two challenges can lead to detecting spurious causal links among variables of interest. In this paper, we study the problem of causal discovery in systems where these two challenges can be present simultaneously. We consider linear models which include four types of variables: variables that are directly observed, variables that are not directly observed but are measured with error, the corresponding measurements, and variables that are neither observed nor measured. We characterize the extent of identifiability of such model under separability condition (i.e., the matrix indicating the independent exogenous noise terms pertaining to the observed variables is identifiable) together with two versions of faithfulness assumptions and propose a notion of observational equivalence. We provide graphical characterization of the models that are equivalent and present a recovery algorithm that could return models equivalent to the ground truth.

• The paper introduces an LV-SEM-ME framework that integrates measurement error with latent variable analysis for improved causal discovery.
• The paper rigorously investigates model identifiability using both conventional and specialized LV-SEM-ME faithfulness assumptions.
• The paper defines AOG and DOG equivalence classes and proposes recovery algorithms to extract causal structures from observational data.

Causal Discovery in Linear Models with Unobserved Variables and Measurement Error

The paper by Yang et al. addresses the intricate problem of causal discovery in linear models where both unobserved variables and measurement error are present. This dual challenge is highly relevant for practical applications in many domains, but it has been inadequately addressed in previous literature. The paper introduces a specific type of model, termed Linear Latent Variable Structural Equation Models with Measurement Error (LV-SEM-ME), and offers substantial advancements in understanding and addressing these issues.

Key Contributions

1. Model Formulation: The paper defines a framework for LV-SEM-ME, which includes four types of variables: directly observed variables, measured variables with error, unobserved variables, and the corresponding measurements. This model accounts for both measurement error and unobserved common causes, differing from previous models that typically handle these challenges separately.
2. Identifiability Analysis: The paper thoroughly investigates the identifiability of LV-SEM-ME models. It employs two variants of faithfulness assumptions: (1) a conventional faithfulness assumption regarding zero causal effects, and (2) a more intricate LV-SEM-ME faithfulness that addresses additional parameter proportionalities. Through this, the authors delineate the extent to which LV-SEM-ME models can be identified.
3. Equivalence Classes: The concept of Ancestral Ordered Grouping (AOG) and Direct Ordered Grouping (DOG) equivalence classes are introduced. These classes provide a novel graphical characterization of the models, capturing their causal structures and allowing for the identification of equivalence among models under different assumptions.
4. Algorithmic Solutions: The authors propose algorithms for recovering LV-SEM-ME models, detailing how these algorithms can return models within the AOG and DOG equivalence classes. This is significant for practical applications where model recovery from observational data is crucial.

Implications and Future Directions

The practical implications of this work are considerable. The ability to reliably discover causal structures in the presence of both unobserved components and measurement errors will advance fields such as epidemiology, social sciences, and genomics, where these issues frequently occur. The paper also opens several avenues for future research:

• Improved Estimation Techniques: As the identification of LV-SEM-ME models heavily relies on the accurate estimation of the mixing matrix, further work could focus on more robust or efficient algorithms for mixing matrix recovery, potentially exploring non-Gaussian or overcomplete ICA contexts.
• Non-linear Models: Extending these methods to handle non-linear causal models could broaden their applicability and offer insights into more complex systems.
• Handling Real-World Data: Future studies could implement and test these algorithms on real-world datasets to validate the theoretical advancements and understand any additional practical challenges that might arise in specific application domains.

Conclusion

The contributions of this paper provide a refined understanding of causal discovery in settings complicated by measurement error and latent variables. By framing the problem in terms of LV-SEM-ME models and establishing identifiability conditions through AOG and DOG equivalence classes, the authors offer a comprehensive framework and practical solutions that enhance our ability to derive causal insights from complex systems. This work represents a significant step forward in the field of causal inference and structural equation modeling.