Iterative Identification Closure: Amplifying Causal Identifiability in Linear SEMs

Ziyi Ding; Xiao-Ping Zhang

stat.MLApr 10, 2026

Iterative Identification Closure: Amplifying Causal Identifiability in Linear SEMs

Authors: Ziyi Ding, Xiao-Ping Zhang

Organizations: Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen, China

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.

Figures & tables

Figure 1: Overview of Iterative Identification Closure (IIC). (a) Framework: diverse seed functions provide initial identifiable edges $\mathcal{S}_{0}$ ; 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 $|\mathrm{pa}(i)|$ , enabling Reduced HTC to resolve further edges until fixed-point convergence. (b) Core mechanism on a 4-node example: HTC fails for $Y$ ( $T\in\mathrm{pa}(Y)\cap\mathrm{sib}(Y)$ ); IV seed identifies $Z\to T$ ; after substituting known $B_{ZT}$ , Reduced HTC only needs $|R|=1$ remaining parent and successfully identifies $T\to Y$ and $W\to Y$ . (c) IIC achieves 97.5% identification (propagation gain $\gamma\approx 4\times$ ) vs. 77–85% for HTC alone, with zero false positives on 134,144 edges.

Panel of Figure 2 — Figure 2: Iterative propagation of IIC (5-node example). Green = identified; Yellow = newly identified this round. $t=0$ : IV seed identifies $Z\to T$ and $Z\to U$ . $t=1$ : 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 $T,W\in\mathrm{sib}(Y)$ ).

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)

Panel of Figure 3 — 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.

Panel of Figure 4 — 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 $\gamma=|\mathrm{IIC}(\mathcal{S}_{0})\setminus\text{HTC}|/|\mathcal{S}_{0}|$ : IIC achieves up to $4.0\times$ amplification, far exceeding prior methods ( $\gamma\leq 1.2$ ).

Method	Id. Rate $\uparrow$	vs. HTC $\uparrow$	Unique gains $\uparrow$
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)

Panel of Figure 5 — 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 .

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

$\|R\|$	$\|R\cap\mathrm{sib}(i)\|$	Gap edges
2	1	2,784 (78.7%)
2	2	544 (15.4%)
3	1	176 (5.0%)
3	2	28 (0.8%)
3	3	4 (0.1%)

Appendix

Table 4: Algebraic structure of IIC-gap edges (

n=5

, exhaustive)

$n$	Both	IIC Only	AD Only	Neither
4	288 (85.7%)	10 (3.0%)	0	38 (11.3%)
5	108,368 (80.8%)	2,080 (1.6%)	0	23,696 (17.7%)

Appendix

Table 5: IIC (IV seed) vs. Ancestor Decomposition: exhaustive comparison

$n$	Graphs	Mean Iter	Max Iter	$\leq 2$ Iter
4	96	1.83	2	100%
5	24,064	1.97	2	100%
6 (random)	982	1.96	2	100%
7 (random)	500	1.99	2	100%

Appendix

Table 6: IIC convergence speed

$n$	Newly Id	True Pos.	False Pos.	Precision
4	10	10	0	100%
5	2,080	2,080	0	100%

Appendix

Table 7: IIC newly identified edges vs. numerical Jacobian ground truth (IV-structured graphs)

Condition	Graphs	Edges	Gap	Complete?
All graphs	24,064	134,144	17.1%	NO
$\mathrm{pa}(i)\cap\mathrm{sib}(i)=\emptyset\;\forall i$	2,142	10,374	0.0%	YES
$\|\mathrm{sib}(i)\|=0\;\forall i$	376	2,096	0.0%	YES
Tree-like ( $\|\mathrm{pa}(i)\|\leq 1$ )	1,600	5,760	0.0%	YES
$\|\mathrm{sib}(i)\|\leq 1\;\forall i$	3,760	20,960	4.9%	NO
$\max\|\mathrm{pa}(i)\|\leq 2$	15,808	80,384	9.4%	NO

Appendix

Table 8: Graph class conditions and IIC gap (

n=5

, exhaustive verification)

$n$	$\|E\|$	$\|S_{0}\|$	HTC%	IIC%	Gain	Time (ms)
10	14	2	96.1 $\pm$ 2.3	98.6 $\pm$ 1.1	+2.5	18 $\pm$ 5
20	57	4	97.0 $\pm$ 1.8	99.2 $\pm$ 0.7	+2.2	72 $\pm$ 12
50	368	10	98.6 $\pm$ 0.9	99.3 $\pm$ 0.5	+0.7	580 $\pm$ 45
100	1,485	20	99.5 $\pm$ 0.3	99.7 $\pm$ 0.2	+0.1	5,595 $\pm$ 310

Appendix

Table 9: Scalability of IIC on large random graphs (intervention seed,

k=n/5

nodes, 200 graphs

\pm

std)

Edge	Method	$\|\text{Bias}\|$ $\downarrow$	RMSE $\downarrow$
$Z\to T$ (no conf.)	IIC	0.003 $\pm$ 0.016	0.017 $\pm$ 0.003
	2SLS	0.003 $\pm$ 0.016	0.017 $\pm$ 0.003
	OLS	0.002 $\pm$ 0.015	0.015 $\pm$ 0.003
$T\to Y$ (conf.)	IIC	0.001 $\pm$ 0.019	0.020 $\pm$ 0.004
	2SLS	0.001 $\pm$ 0.019	0.020 $\pm$ 0.004
	OLS	0.213 $\pm$ 0.015	0.213 $\pm$ 0.003

Appendix

Table 10: IIC-Estimate vs. 2SLS vs. OLS (6-node graph,

n=5000

samples, averaged over 100 trials

\pm

std). Bold = best method per edge group.

Panel of Figure 6 — Figure 6: Box plot of per-trial absolute estimation error ( $|\hat{B}_{ji}-B^{*}_{ji}|$ , 100 trials, $n=5000$ ). On confounded edges ( $T{\to}Y$ , $W_{1}{\to}Y$ , $W_{2}{\to}Y$ ), OLS exhibits large systematic bias; IIC achieves near-zero error. 2SLS matches IIC where an IV exists but cannot estimate edges without a valid IV (marked N/A).

$k$	$n\!=\!10$ Rate	Gain	$n\!=\!20$ Rate	Gain
0	95.5%	—	96.5%	—
1	97.2%	+1.7%	97.3%	+0.9%
2	98.6%	+3.1%	98.2%	+1.7%
3	99.1%	+3.6%	98.5%	+2.1%
5	99.7%	+4.2%	99.2%	+2.7%
7	100.0%	+4.4%	99.6%	+3.1%

Appendix

Table 11: Number of intervention nodes

k

vs. identification rate (random graphs, mean

\pm

std)

Panel of Figure 7 — Figure 7: Identification rate vs. number of intervention nodes. The first 2 nodes yield $\sim$ 60–70% of the total gain.

Perturbation	Rate	Precision	Recall	Id. Rate
None (correct)	0%	1.000	1.000	0.925
Missing directed	10%	0.994	0.987	0.918
	20%	0.994	0.987	0.918
	30%	0.991	0.983	0.917
Extra directed	10%	0.991	0.981	0.914
	20%	0.991	0.980	0.913

Appendix

Table 12: IIC robustness under graph misspecification (

n=6

, intervention

k=1

)

	RMSE			Coverage (95%)
$n$	$B_{01}$	$B_{12}$	$B_{31}$	$B_{01}$	$B_{12}$	$B_{31}$
100	0.116	0.160	0.096	95.5%	97.0%	94.5%
500	0.056	0.068	0.046	94.0%	96.0%	94.5%
2,000	0.027	0.033	0.025	94.5%	94.0%	93.5%
10,000	0.012	0.015	0.010	96.5%	93.5%	97.0%

Appendix

Table 13: IIC-IV estimation: 5-node graph, 3/5 edges identifiable (

B_{01}^{*}=0.8

B_{12}^{*}=-0.5

B_{31}^{*}=0.6

)

Panel of Figure 8 — Figure 8: Convergence of IIC-Estimate RMSE with sample size. The RMSE of all three edges decreases along the $O(1/\sqrt{n})$ reference line, validating the $\sqrt{n}$ -consistency of Theorem D.1 .

$n$	Mean $\|\text{Bias}\|$	RMSE
200	0.157	0.280
1,000	0.057	0.081
5,000	0.027	0.038

Appendix

Table 14: IIC-Intervention estimation: 6-node graph, 2 nodes intervened, 7/7 edges identifiable

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Sign Identifiability of Causal Effects in Stationary Stochastic Dynamical Systems

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1LIACS, Leiden University, Netherlands

Stochastic Differential Equations Structural Causal Models Causal Identifiability

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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.

Abstract

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