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.
Figures & tables
Figure 1: Overview of Iterative Identification Closure (IIC). (a) Framework: diverse seed functions provide initial identifiable edges S0 ; Standard HTC and the novel Reduced HTC operate in parallel to expand the identified set; the red dashed arrow represents iterative feedback—newly identified edges feed back to reduce ∣pa(i)∣ , enabling Reduced HTC to resolve further edges until fixed-point convergence. (b) Core mechanism on a 4-node example: HTC fails for Y ( T∈pa(Y)∩sib(Y) ); IV seed identifies Z→T ; after substituting known BZT , Reduced HTC only needs ∣R∣=1 remaining parent and successfully identifies T→Y and W→Y . (c) IIC achieves 97.5% identification (propagation gain γ≈4× ) vs. 77–85% for HTC alone, with zero false positives on 134,144 edges.
Figure 2: Iterative propagation of IIC (5-node example). Green = identified; Yellow = newly identified this round. t=0 : IV seed identifies Z→T and Z→U . t=1 : After substituting known edges, all 3 incoming edges of Y satisfy Reduced HTC. t=2 : All edges identified; fixed point reached. Standard HTC cannot identify any incoming edge of Y (since T,W∈sib(Y) ).
Seed Source
Id Rate
Gain vs. HTC
Total Edges
No seed (= HTC)
85.6%
—
10,283
Intervention ( k=1 )
93.4%
+7.8%
10,283
Intervention ( k=2 )
97.5%
+11.9%
10,283
Table 1: IIC with intervention seeds on general random graphs ( n=6 , 1881 graphs)
Seed Source
n=4 Id Rate
n=4 Gap
n=5 Id Rate
n=5 Gap
No seed (= HTC)
85.7%
14.3%
80.8%
19.2%
IV seed
88.7%
11.3%
82.3%
17.7%
Exogenous seed
85.7%
14.3%
80.8%
19.2%
IV + Exogenous
88.7%
11.3%
82.3%
17.7%
Table 2: IIC identification rates under different seed sources (IV-structured graphs, exhaustive enumeration)
Figure 3: HTC vs. IIC identification rate (left axis, solid lines) and IIC runtime (right axis, dashed line) on log-scale x -axis. IIC consistently outperforms HTC across all graph sizes, with runtime <6 seconds for 100-node graphs.
Figure 4: Identification amplification analysis. (a) Breakdown of edge classification: HTC baseline (blue), IIC additional gains via Reduced HTC propagation (orange), and remaining gap (red). Combined seeds (IV+intervention) reduce the gap from 17% to 2.6%. (b) Propagation gain γ=∣IIC(S0)∖HTC∣/∣S0∣ : IIC achieves up to 4.0× amplification, far exceeding prior methods ( γ≤1.2 ).
Method
Id. Rate ↑
vs. HTC ↑
Unique gains ↑
HTC (baseline)
85.6%
—
—
Chen et al. [10]
93.1%
+7.5%
104
Xie et al. [13]
86.8%
+1.3%
15
IIC (interv k=2 )
97.5%
+11.9%
554
Table 3: Identification rates on random mixed graphs ( n=6 , 1881 graphs, 10283 edges)
Figure 5: MR network for CHD (9 nodes, 13 directed, 4 bidirected edges). HTC gap: 5/13 edges (all into CHD). IIC with IV seeds: 13/13 identified (100%). Details in Appendix C.10 .
Figure 8: Convergence of IIC-Estimate RMSE with sample size. The RMSE of all three edges decreases along the O(1/n) reference line, validating the n -consistency of Theorem D.1 .
We study identifiability in continuous-time linear stationary stochastic differential equations with a known causal structure. Unlike existing approaches, we relax the assumption of a known diffusion matrix, thereby respecting the model's intrinsic scale invariance. Therefore, rather than recovering drift coefficients themselves, we introduce edge-sign identifiability: for a given causal structure, we ask whether the sign of a given drift entry is uniquely determined across all observational covariance matrices induced by parametrisations compatible with that structure. This leads to a trichotomy of edge-sign identifiability: identifiable, non-identifiable, and partially identifiable. This trichotomy introduces the new notion of partial identifiability to the literature, which we show is a genuine category in our setting. Under a notion of faithfulness, we derive criteria to identify membership of each category for general graphs. Applying our criteria to specific causal structures, both analogous to classical causal settings (e.g., instrumental variables) and novel cyclic settings, we determine their edge-sign identifiability and, in some cases, obtain explicit expressions for the sign of a target edge in terms of the observational covariance matrix.
Continuous-time stochastic differential equations (SDEs) provide a natural framework for modeling interacting dynamical systems. A central causal question is whether causal effects among observed variables can be identified from observational data drawn from the stationary distribution, given the underlying causal graph. For linear SDEs, existing identifiability results are largely restricted to fully observed systems whose causal graphs contain no directed 2-cycles. We first establish new graphical conditions for the generic identifiability of causal effects in systems that may contain directed 2-cycles, thereby allowing for reciprocal interactions. Next, we address the previously unexplored setting of latent variables by developing an instrumental-variable framework for identifying a target direct causal effect under partial observability.
Ana Leticia Garcez Vicente, Saber Salehkaleybar
Leiden Institute of Advanced Computer Science (LIACS), Leiden University, Netherlands
Determining identifiability of causal effects from observational data under latent confounding is a central challenge in causal inference. For linear structural causal models, identifiability of causal effects is decidable through symbolic computation. However, standard approaches based on Gröbner bases become computationally infeasible beyond small settings due to their doubly exponential complexity. In this work, we study how to practically use symbolic computation for deciding rational identifiability. In particular, we present an efficient algorithm that provably finds the lowest degree identifying formulas. For a causal effect of interest, if there exists an identification formula of a prespecified maximal degree, our algorithm returns such a formula in quasi-polynomial time.
Benjamin Hollering, Pratik Misra, Nils Sturma
Max Planck Institute for Mathematics in the Sciences, Germany · Binghamton University, USA · École Polytechnique Fédérale de Lausanne, Switzerland