Iterative Identification Closure: Amplifying Causal Identifiability in Linear SEMs
2026-04-10 • Machine Learning
Machine Learning
AI summaryⓘ
The authors discuss a new method called Iterative Identification Closure (IIC) that improves identifying cause-effect relationships in complex models with hidden factors. Unlike the common Half-Trek Criterion (HTC) that looks at one node at a time, IIC uses an iterative process where newly found connections help find even more. They prove their approach is reliable, works quickly, and is better than existing methods, closing most of the gaps left by HTC. Their tests on small graphs show perfect accuracy and much higher identification rates when extra information is used.
Half-Trek Criterioncausal identificationstructural equation modelslatent confoundersinstrumental variablesJacobian matrixcovariance structuregraph theorygeneric identifiabilityiterative algorithms
Authors
Ziyi Ding, Xiao-Ping Zhang
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.