Iterative Identification Closure: Amplifying Causal Identifiability in Linear SEMs
| Authors | Ziyi Ding & Xiao-Ping Zhang |
| Year | 2026 |
| Field | Statistics / ML |
| arXiv | 2604.09309 |
| Download | |
| Categories | stat.ML, cs.LG, stat.CO |
Abstract
The Half-Trek Criterion (HTC) is the primary graphical tool for determining generic identifiability of causal effect coefficients in linear structural equation models (SEMs) with latent confounders. However, HTC is inherently node-wise: it simultaneously resolves all incoming edges of a node, leaving a gap of "inconclusive" causal effects (15-23% in moderate graphs). We introduce Iterative Identification Closure (IIC), a general framework that decouples causal identification into two phases: (1) a seed function S_0 that identifies an initial set of edges from any external source of information (instrumental variables, interventions, non-Gaussianity, prior knowledge, etc.); and (2) Reduced HTC propagation that iteratively substitutes known coefficients to reduce system dimension, enabling identification of edges that standard HTC cannot resolve. The core novelty is iterative identification propagation: newly identified edges feed back to unlock further identification -- a mechanism absent from all existing graphical criteria, which treat each edge (or node) in isolation. This propagation is non-trivial: coefficient substitution alters the covariance structure, and soundness requires proving that the modified Jacobian retains generic full rank -- a new theoretical result (Reduced HTC Theorem). We prove that IIC is sound, monotone, converges in O(|E|) iterations (empirically <=2), and strictly subsumes both HTC and ancestor decomposition. Exhaustive verification on all graphs with n<=5 (134,144 edges) confirms 100% precision (zero false positives); with combined seeds, IIC reduces the HTC gap by over 80%. The propagation gain is gamma~4x (2 seeds identifying ~3% of edges to 97.5% total identification), far exceeding gamma<=1.2x of prior methods that incorporate side information without iterative feedback.
Engineering Breakdown
Plain English
This paper solves a key limitation in causal inference for linear structural equation models (SEMs) with hidden confounders. The Half-Trek Criterion (HTC), the standard tool for identifying causal effects, leaves 15-23% of relationships inconclusive in moderate-sized graphs. The authors introduce Iterative Identification Closure (IIC), which combines any initial source of causal knowledge (instruments, interventions, non-Gaussianity, priors) with iterative graph reduction to identify significantly more causal effects that HTC alone cannot resolve.
Key Engineering Insight
IIC decouples causal identification into a two-phase process: seed known effects from any available information source, then propagate that knowledge iteratively by substituting solved coefficients back into the system to reduce its dimensionality. This modular design lets engineers plug in domain-specific identification strategies without reimplementing the core algorithm.
Why It Matters for Engineers
In production causal inference systems, unresolved causal effects mean incomplete models and lower confidence in root-cause analysis or counterfactual predictions. A 15-23% failure rate directly translates to gaps in decision support systems, especially in finance, healthcare, and operations. IIC's ability to recover these missing relationships using any available signal (measurement instruments, A/B tests, data distributions) makes causal inference more practical where perfect experimental data doesn't exist.
Research Context
HTC has been the standard graphical criterion for causal identifiability in SEMs since its introduction, but it's fundamentally node-wise—it tries to resolve all edges at once and fails when the graph structure admits ambiguity. This paper advances the field by showing that sequential, iterative approaches can recover identifiability that simultaneous methods miss. It also generalizes the framework to integrate multiple identification sources, bridging the gap between pure graphical methods and practical hybrid approaches used in industry.
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