[Paper Review] Decomposition and Identification of Linear Structural Equation Models

Bryant Chen|arXiv (Cornell University)|Aug 7, 2015

Bayesian Modeling and Causal Inference4 citations

TL;DR

This paper extends the half-trek criterion to identify structural coefficients in linear structural equation models (SEMs), introducing recursive c-component decomposition to enhance identification power. It demonstrates that the proposed method subsumes non-parametric identification algorithms for linear models, enabling identification of coefficients previously undetectable by existing linear methods.

ABSTRACT

In this paper, we address the problem of identifying linear structural equation models. We first extend the edge set half-trek criterion to cover a broader class of models. We then show that any semi-Markovian linear model can be recursively decomposed into simpler sub-models, resulting in improved identification power. Finally, we show that, unlike the existing methods developed for linear models, the resulting method subsumes the identification algorithm of non-parametric models.

Motivation & Objective

• Address the limitation of existing linear SEM identification methods that fail to identify certain coefficients identifiable via non-parametric approaches.
• Develop a method that identifies structural coefficients directly from the population covariance matrix, avoiding reliance on iterative estimation software.
• Improve identification power in semi-Markovian linear models by recursively decomposing the model into simpler sub-models using c-components.
• Demonstrate that the proposed method subsumes non-parametric identification algorithms when applied to linear SEMs, resolving a key gap in prior work.

Proposed method

• Extend the edge set half-trek criterion to handle a broader class of models, including non-Markovian structures.
• Apply recursive c-component decomposition to simplify the model into sub-models, enabling identification of coefficients that are otherwise unidentifiable.
• Use generalized half-trek criterion (g-HTC) to identify coefficients in decomposed sub-models by leveraging d-separation and conditional independence in the covariance matrix.
• Implement a recursive algorithm that alternately decomposes the graph into c-components and removes descendant sets to expose new identification opportunities.
• Integrate identification in sub-models with re-identification in the original graph by un-removing marginalized nodes when new coefficients become identifiable.
• Leverage the fact that if a direct effect is identifiable in a non-parametric model, then the corresponding coefficient is also identifiable in the linear model using the proposed method.

Experimental results

Research questions

• RQ1 Can the half-trek criterion be extended to identify more coefficients in linear SEMs, particularly in non-Markovian models?
• RQ2 Does recursive c-component decomposition improve identification power in semi-Markovian linear SEMs?
• RQ3 Can the proposed method identify coefficients that are identifiable via non-parametric methods but not through existing linear SEM identification algorithms?
• RQ4 Is there a systematic way to exploit decomposition and g-HTC to identify coefficients that remain unidentified after standard linear identification procedures?

Key findings

• The extended half-trek criterion successfully identifies coefficients in a broader class of linear SEMs, including non-Markovian models.
• Recursive c-component decomposition enables the identification of additional coefficients by simplifying the model structure and revealing conditional independence relationships.
• The method subsumes non-parametric identification algorithms for linear SEMs, meaning any coefficient identifiable via non-parametric methods is also identifiable using this approach.
• The algorithm can identify coefficients even when the full model is not globally identified, allowing partial identification of structural parameters prior to data collection.
• The recursive decomposition process may need to be repeated across multiple iterations of decomposition and re-identification to fully exploit identification potential.
• The method identifies the coefficient $ h $ in Figure 2(a), which was previously undetected by standard linear identification algorithms, despite being identifiable via non-parametric means.