Causal Discovery Under Hard Selection Bias: A New Robust Score-Matching Approach

Yiwen Qiu; Francesco Montagna; Shimeng Huang; Francesco Locatello

cs.LGOct 13, 2026

Causal Discovery Under Hard Selection Bias: A New Robust Score-Matching Approach

Authors: Yiwen Qiu, Francesco Montagna, Shimeng Huang, Francesco Locatello

Organizations: IST Austria, Am Campus 1, Klosterneuburg, 3400

Abstract

Selection bias is a difficult yet widespread problem in causal discovery, occurring whenever non-random selection processes lead to data that is not representative of the underlying populations. Due to its practical importance, related works have been proposed to address the problem under local [Versteeg et al., 2022] or interventional settings [Dai et al., 2025a], but general algorithms for observational data remain elusive. In this paper, we focus on truncated data (hard selection), and provide a general result for Additive Noise Models (ANM). We show that score-based methods are a surprisingly suitable candidate due to a special property of the score function ( $\nabla \log p(x)$ ): deterministic truncation preserves both the score of the joint distribution as well as the Jacobian of the score. While established score-matching algorithms still fail in most settings, we actively leverage this observation to propose a new method for causal discovery that is robust under truncated data with ANMs. Overall, our approach offers a general, robust solution that is agnostic to external information about the selection process and still achieves comparable performance to the state-of-the-art approaches when selection is not present.

Figures & tables

ANM Assumption	LiNGAM	CAM-UV	SCORE	NoGAM	CHess (Ours)
Nonlinear Gaussian
Linear non-Gaussian
Nonlinear non-Gaussian
Hard Selection (Truncation)			∗

Table 1: Identifiability guarantees of causal discovery methods across ANM assumptions. denotes full DAG identifiability; denotes no guarantee. LiNGAM ( Shimizu et al., 2006 ; Shimizu et al., 2011 ) only deals with linear non-Gaussian ANMs, SCORE ( Rolland et al., 2022 ) requires nonlinear mechanisms with Gaussian noise. NoGAM ( Montagna et al., 2025 ) extends SCORE to arbitrary noise distributions. CAM-UV ( Maeda and Shimizu, 2021 ) additionally handles latent confounders under nonlinear separable (CAM) assumptions. SCORE can handle latent hard selection bias in both Gaussian and non-Gaussian cases, which is not discussed in the original paper but is one of the surprising findings in this work ( ∗ ) . Our method (CHess) is the only method that is resilient to hard selection under all the above ANMs. Linear Gaussian models are known to be not identifiable beyond the Markov equivalence class ( Peters and Bühlmann, 2014 ) thus omitted in the table.

Panel of Figure 1 — Figure 1: In the subgraph, the sink node $X_{c}$ is only d-connected to its own noise $N_{c}$ after conditioning on $X_{\PA_{c}}$ , while $X_{p}$ is d-connected to $N_{p}$ and $N_{c}$ after conditioning on $X_{\PA_{p}}\cup X_{\CH_{p}}$ .

Setting	Sink rank	Non-sink rank	Gap
Linear Gaussian	$0$	$0$	$0$ (not identifiable)
Linear non-Gaussian	$1$	$\geq 2$	$\geq 1$
Nonlinear Gaussian	$0$	$\geq 1$	$\geq 1$
Nonlinear non-Gaussian	$1$	$\geq 2$	$\geq 1$

Table 2: Rank of

M_{i}^{\mathrm{cond}}{\color[rgb]{0,0,0}(z)}

for sink vs. non-sink nodes across different ANM settings.

	$d=5$ , gumbel			$d=10$ , gumbel
fraction	$\rho=1.0$	$\rho=0.1$	$\rho=0.05$	$\rho=1.0$	$\rho=0.1$	$\rho=0.05$
SCORE	$0.57$ ( $\pm$ 0.25)	$0.37$ ( $\pm$ 0.17)	$0.41$ ( $\pm$ 0.14)	$0.65$ ( $\pm$ 0.12)	$0.46$ ( $\pm$ 0.11)	$0.45$ ( $\pm$ 0.10)
DAS	$0.57$ ( $\pm$ 0.25)	$0.37$ ( $\pm$ 0.17)	$0.41$ ( $\pm$ 0.14)	$0.66$ ( $\pm$ 0.11)	$0.48$ ( $\pm$ 0.12)	$0.49$ ( $\pm$ 0.10)
NoGAM	$0.81$ ( $\pm$ 0.21)	$0.40$ ( $\pm$ 0.16)	$0.40$ ( $\pm$ 0.14)	$0.78$ ( $\pm$ 0.16)	$0.47$ ( $\pm$ 0.14)	$0.45$ ( $\pm$ 0.11)
CAM	$0.31$ ( $\pm$ 0.20)	$0.18$ ( $\pm$ 0.14)	$0.19$ ( $\pm$ 0.12)	$0.23$ ( $\pm$ 0.08)	$0.22$ ( $\pm$ 0.11)	$0.19$ ( $\pm$ 0.12)
DirectLiNGAM	$\mathbf{0.99}$ ( $\pm$ 0.03)	$0.64$ ( $\pm$ 0.14)	$0.65$ ( $\pm$ 0.16)	$\mathbf{0.98}$ ( $\pm$ 0.03)	$0.56$ ( $\pm$ 0.05)	$0.54$ ( $\pm$ 0.07)

Table 3: F1 (Arrow) for Linear-nG (gumbel),

d=5

and

d=10

Method	$\rho=1$ ( $n=373$ )	$\rho=0.1$
SCORE	$0.28$ ( $\pm$ 0.05)	$0.29$ ( $\pm$ 0.05)
DAS	$0.19$ ( $\pm$ 0.10)	$0.23$ ( $\pm$ 0.08)
NoGAM	$0.29$ ( $\pm$ 0.04)	$0.30$ ( $\pm$ 0.06)
CAM	$0.40$ ( $\pm$ 0.08)	$0.28$ ( $\pm$ 0.09)
PC	$0.25$ ( $\pm$ 0.09)	$0.23$ ( $\pm$ 0.07)
FCI	$0.13$ ( $\pm$ 0.04)	$0.14$ ( $\pm$ 0.11)

Table 4: F1 (Arrow) on Sachs Dataset for

\rho=1

(downasmpled) and

\rho=0.1

Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

Setting	sink rank	Non-sink rank	Gap
Linear + Gaussian	$1$	$1$	$0$ (not identifiable)
Linear + non-Gaussian	$1$	$\geq 1+\|\CH_{l}\|$	$\|\CH_{j}\|\geq 1$
Nonlinear + Gaussian	$\leq\|\PA_{l}\|+1$	$\leq\|\PA_{l}\|+\|\CH_{l}\|+1$	$\|\PA_{j}\|+\|\CH_{j}\|-\|\PA_{l}\|$
Nonlinear + non-Gaussian	$\leq\|\PA_{l}\|+1$	$\leq\|\PA_{l}\|+\|\CH_{l}\|+1$	$\|\PA_{j}\|+\|\CH_{j}\|-\|\PA_{l}\|$

Appendix

Table 5: Rank of

M_{j}

for sink vs. non-sink nodes across different ANM settings.

True	Estimated	Adjacency	Arrowhead
$a\leftarrow b$	$a\leftarrow b$	TP	TP, TN
	$a\to b$	TP	FP, FN
	$a\text{ --- }b$	TP	FN
	$a\quad\ \ b$	FN	FN
$a\quad\ \ b$	$a\leftarrow b$	FP	FP
	$a\to b$	FP	FP

Appendix

Table 6: Metrics used for evaluation. Precision=TP/(TP+FP), Recall=TP/(TP+FN).

	F1 (Adj) $\uparrow$			F1 (Arrow) $\uparrow$
Method	$K=0$	$K=1$	$K=2$	$K=0$	$K=1$	$K=2$
PC	$\mathbf{0.96}(0.00)$	$0.76(0.02)$	$0.70(0.02)$	$\backslash$	$\backslash$	$\backslash$
FCI	$\mathbf{0.96}(0.00)$	$0.76(0.02)$	$0.70(0.02)$	$\backslash$	$\backslash$	$\backslash$
Exact	$0.70(0.02)$	$0.65(0.02)$	$0.66(0.01)$	$0.29(0.04)$	$0.31(0.02)$	$0.34(0.06)$
GES	$0.96(0.00)$	$\underline{0.78}(0.02)$	$0.72(0.02)$	$\backslash$	$\backslash$	$\backslash$
GRaSP	$0.89(0.01)$	$\mathbf{0.79}(0.03)$	$0.71(0.02)$	$\backslash$	$\backslash$	$\backslash$

Appendix

Table 7:

d=5

, noise=lognormal: F1 Metrics

	SHD $\downarrow$
Method	$K=0$	$K=1$	$K=2$
PC	$2.81(1.85)$	$4.81(3.26)$	$5.11(2.25)$
FCI	$4.48(0.92)$	$5.96(0.85)$	$6.33(2.07)$
Exact	$6.11(2.84)$	$6.33(2.59)$	$6.22(4.02)$
GES	$2.78(1.58)$	$4.63(3.79)$	$5.70(3.54)$
GRaSP	$3.44(1.80)$	$4.00(2.74)$	$5.15(2.13)$

Appendix

Table 8:

d=5

, noise=lognormal: SHD Metrics

	F1 (Adj) $\uparrow$			F1 (Arrow) $\uparrow$
Method	$K=0$	$K=1$	$K=2$	$K=0$	$K=1$	$K=2$
PC	$0.90(0.01)$	$0.76(0.01)$	$0.73(0.01)$	$0.59(0.04)$	$\backslash$	$\backslash$
FCI	$0.90(0.01)$	$0.76(0.01)$	$0.73(0.01)$	$0.55(0.03)$	$\backslash$	$\backslash$
Exact	$0.53(0.00)$	$0.50(0.00)$	$0.46(0.00)$	$0.23(0.01)$	$0.25(0.01)$	$0.21(0.01)$
GES	$0.89(0.00)$	$0.70(0.01)$	$0.64(0.01)$	$0.58(0.04)$	$0.49(0.02)$	$0.44(0.02)$
GRaSP	$0.88(0.01)$	$\mathbf{0.80}(0.01)$	$\mathbf{0.75}(0.02)$	$0.55(0.05)$	$\backslash$	$\backslash$

Appendix

Table 9:

d=10

, noise=lognormal: F1 Metrics

	SHD $\downarrow$
Method	$K=0$	$K=1$	$K=2$
PC	$5.89(4.84)$	$\underline{7.78}(6.10)$	$\underline{9.85}(7.02)$
FCI	$9.74(1.97)$	$11.85(3.61)$	$12.85(5.31)$
Exact	$20.56(6.17)$	$21.74(11.16)$	$24.56(6.32)$
GES	$6.11(4.40)$	$10.33(7.19)$	$13.33(20.44)$
GRaSP	$5.96(5.07)$	$\mathbf{7.22}(6.84)$	$\mathbf{9.15}(15.16)$

Appendix

Table 10:

d=10

, noise=lognormal: SHD Metrics

	F1 (Adj) $\uparrow$			F1 (Arrow) $\uparrow$
Method	$K=0$	$K=1$	$K=2$	$K=0$	$K=1$	$K=2$
PC	$\underline{0.98}(0.00)$	$0.75(0.01)$	$0.67(0.01)$	$\backslash$	$0.41(0.05)$	$\backslash$
FCI	$\underline{0.98}(0.00)$	$0.75(0.01)$	$0.66(0.02)$	$\backslash$	$0.41(0.05)$	$\backslash$
Exact	$0.76(0.01)$	$0.71(0.01)$	$0.68(0.01)$	$0.30(0.03)$	$0.31(0.03)$	$0.36(0.03)$
GES	$0.96(0.00)$	$0.71(0.01)$	$0.65(0.01)$	$\backslash$	$\backslash$	$\backslash$
GRaSP	$0.95(0.00)$	$\mathbf{0.76}(0.01)$	$0.66(0.02)$	$\backslash$	$\backslash$	$\backslash$

Appendix

Table 11:

d=5

, noise=gumbel: F1 Metrics

	SHD $\downarrow$
Method	$K=0$	$K=1$	$K=2$
PC	$2.37(1.94)$	$5.00(3.11)$	$5.93(1.55)$
FCI	$4.19(1.04)$	$6.63(1.64)$	$7.19(1.04)$
Exact	$5.74(1.75)$	$5.81(2.00)$	$5.81(1.85)$
GES	$2.59(2.32)$	$5.89(1.73)$	$6.33(1.85)$
GRaSP	$2.78(2.10)$	$4.85(2.50)$	$6.04(2.85)$

Appendix

Table 12:

d=5

, noise=gumbel: SHD Metrics

	F1 (Adj) $\uparrow$			F1 (Arrow) $\uparrow$
Method	$K=0$	$K=1$	$K=2$	$K=0$	$K=1$	$K=2$
PC	$0.96(0.00)$	$0.77(0.01)$	$0.66(0.01)$	$\backslash$	$0.54(0.02)$	$0.47(0.02)$
FCI	$0.96(0.00)$	$\underline{0.77}(0.01)$	$\underline{0.67}(0.01)$	$\backslash$	$0.46(0.01)$	$0.37(0.01)$
Exact	$0.57(0.00)$	$0.53(0.00)$	$0.52(0.00)$	$0.19(0.01)$	$0.25(0.01)$	$0.27(0.02)$
GES	$0.96(0.00)$	$0.64(0.00)$	$0.55(0.01)$	$0.64(0.06)$	$0.44(0.02)$	$0.32(0.01)$
GRaSP	$\underline{0.97}(0.00)$	$\mathbf{0.81}(0.01)$	$\mathbf{0.69}(0.01)$	$0.65(0.05)$	$0.45(0.05)$	$0.34(0.03)$

Appendix

Table 13:

d=10

, noise=gumbel: F1 Metrics

	SHD $\downarrow$
Method	$K=0$	$K=1$	$K=2$
PC	$4.96(5.29)$	$\underline{9.37}(6.60)$	$\mathbf{12.26}(10.64)$
FCI	$9.30(2.43)$	$13.67(9.56)$	$16.15(10.13)$
Exact	$20.26(6.19)$	$21.48(7.95)$	$20.96(11.89)$
GES	$5.30(9.84)$	$14.78(9.14)$	$19.48(20.18)$
GRaSP	$4.81(5.56)$	$\mathbf{8.96}(7.22)$	$\underline{12.56}(11.73)$

Appendix

Table 14:

d=10

, noise=gumbel: SHD Metrics

	F1 (Adj) $\uparrow$			F1 (Arrow) $\uparrow$
Method	$K=0$	$K=1$	$K=2$	$K=0$	$K=1$	$K=2$
PC	$0.79(0.02)$	$0.67(0.02)$	$0.64(0.02)$	$\backslash$	$\backslash$	$\backslash$
FCI	$0.79(0.02)$	$0.67(0.02)$	$0.64(0.02)$	$\backslash$	$\backslash$	$\backslash$
Exact	$0.62(0.02)$	$0.62(0.02)$	$0.65(0.01)$	$0.33(0.05)$	$0.32(0.03)$	$0.34(0.03)$
GES	$0.77(0.01)$	$0.67(0.03)$	$0.66(0.02)$	$\backslash$	$\backslash$	$\backslash$
GRaSP	$0.68(0.04)$	$0.65(0.03)$	$0.62(0.02)$	$\backslash$	$\backslash$	$\backslash$

Appendix

Table 15:

d=5

, noise=lognormal, MLP: F1 Metrics

	SHD $\downarrow$
Method	$K=0$	$K=1$	$K=2$
PC	$3.77(1.85)$	$5.67(2.82)$	$5.83(2.47)$
FCI	$5.70(0.74)$	$6.73(1.60)$	$6.93(1.86)$
Exact	$6.23(2.91)$	$6.53(2.78)$	$6.33(1.89)$
GES	$4.27(2.46)$	$5.63(3.37)$	$6.50(2.98)$
GRaSP	$4.43(2.58)$	$5.20(2.49)$	$5.60(1.71)$

Appendix

Table 16:

d=5

, noise=lognormal, MLP: SHD Metrics

Method	Network 1	Network 2	Network 3	Network 4	Network 5
SCORE	$12.0$	$12.0$	$\mathbf{12.0}$	$13.0$	$9.0$
DAS	$12.0$	$13.0$	$13.0$	$13.0$	$\mathbf{8.0}$
NoGAM	$13.0$	$13.0$	$13.0$	$12.0$	$10.0$
CAM	$12.0$	$11.0$	$14.0$	$12.0$	$10.0$
CHess (stein)	$\mathbf{11.0}$	$14.0$	$13.0$	$12.0$	$13.0$
CHess (diffan)	$12.0$	$\mathbf{9.0}$	$\mathbf{12.0}$	$\mathbf{11.8}$	$10.4$

Appendix

Table 17: SHD Results for

K=0

Across All Networks

Method	$\rho=1$ $n=1493$ / $747$ / $373$	$\rho=0.2$ ( $n=1493$ )	$\rho=0.1$ ( $n=747$ )	$\rho=0.05$ ( $n=373$ )
SCORE	$0.21$ ( $\pm$ 0.04) / $0.26$ ( $\pm$ 0.06) / $0.28$ ( $\pm$ 0.05)	$0.28$ ( $\pm$ 0.04)	$0.29$ ( $\pm$ 0.05)	$0.28$ ( $\pm$ 0.08)
DAS	$0.17$ ( $\pm$ 0.06) / $0.18$ ( $\pm$ 0.10) / $0.19$ ( $\pm$ 0.10)	$0.24$ ( $\pm$ 0.05)	$0.23$ ( $\pm$ 0.08)	$0.20$ ( $\pm$ 0.10)
NoGAM	$0.23$ ( $\pm$ 0.02) / $0.26$ ( $\pm$ 0.01) / $0.29$ ( $\pm$ 0.04)	$0.28$ ( $\pm$ 0.04)	$0.30$ ( $\pm$ 0.06)	$0.31$ ( $\pm$ 0.08)
CAM	$\mathbf{0.41}$ ( $\pm$ 0.06) / $0.42$ ( $\pm$ 0.08) / $0.40$ ( $\pm$ 0.08)	$0.23$ ( $\pm$ 0.08)	$0.28$ ( $\pm$ 0.09)	$0.27$ ( $\pm$ 0.10)
PC	$0.27$ ( $\pm$ 0.10) / $0.26$ ( $\pm$ 0.10) / $0.25$ ( $\pm$ 0.09)	$0.23$ ( $\pm$ 0.07)	$0.23$ ( $\pm$ 0.07)	$0.20$ ( $\pm$ 0.09)
FCI	$0.16$ ( $\pm$ 0.05) / $0.16$ ( $\pm$ 0.06) / $0.13$ ( $\pm$ 0.04)	$0.11$ ( $\pm$ 0.10)	$0.14$ ( $\pm$ 0.11)	$0.11$ ( $\pm$ 0.13)

Appendix

Table 18: F1 (Arrow), halfspace selection

Method	$\rho=1$ $n=1493$ / $747$ / $373$	$\rho=0.2$ ( $n=1493$ )	$\rho=0.1$ ( $n=747$ )	$\rho=0.05$ ( $n=373$ )
SCORE	$0.21$ ( $\pm$ 0.04) / $0.26$ ( $\pm$ 0.06) / $0.28$ ( $\pm$ 0.05)	$0.29$ ( $\pm$ 0.06)	$0.32$ ( $\pm$ 0.07)	$0.34$ ( $\pm$ 0.08)
DAS	$0.17$ ( $\pm$ 0.06) / $0.18$ ( $\pm$ 0.10) / $0.19$ ( $\pm$ 0.10)	$0.26$ ( $\pm$ 0.06)	$0.29$ ( $\pm$ 0.09)	$0.32$ ( $\pm$ 0.08)
NoGAM	$0.23$ ( $\pm$ 0.02) / $0.26$ ( $\pm$ 0.01) / $0.29$ ( $\pm$ 0.04)	$0.28$ ( $\pm$ 0.05)	$0.31$ ( $\pm$ 0.06)	$0.33$ ( $\pm$ 0.09)
CAM	$\mathbf{0.41}$ ( $\pm$ 0.06) / $0.42$ ( $\pm$ 0.08) / $0.40$ ( $\pm$ 0.08)	$0.30$ ( $\pm$ 0.08)	$0.31$ ( $\pm$ 0.13)	$0.26$ ( $\pm$ 0.10)
PC	$0.27$ ( $\pm$ 0.10) / $0.26$ ( $\pm$ 0.10) / $0.25$ ( $\pm$ 0.09)	$0.23$ ( $\pm$ 0.10)	$0.25$ ( $\pm$ 0.07)	$0.23$ ( $\pm$ 0.12)
FCI	$0.16$ ( $\pm$ 0.05) / $0.16$ ( $\pm$ 0.06) / $0.13$ ( $\pm$ 0.04)	$0.11$ ( $\pm$ 0.08)	$0.15$ ( $\pm$ 0.08)	$0.15$ ( $\pm$ 0.11)

Appendix

Table 19: F1 (Arrow), annulus selection

Method	$\rho=1$ $n=1493$ / $747$ / $373$	$\rho=0.2$ ( $n=1493$ )	$\rho=0.1$ ( $n=747$ )	$\rho=0.05$ ( $n=373$ )
SCORE	$0.21$ ( $\pm$ 0.04) / $0.26$ ( $\pm$ 0.06) / $0.28$ ( $\pm$ 0.05)	$0.21$ ( $\pm$ 0.06)	$0.24$ ( $\pm$ 0.06)	$0.21$ ( $\pm$ 0.05)
DAS	$0.17$ ( $\pm$ 0.06) / $0.18$ ( $\pm$ 0.10) / $0.19$ ( $\pm$ 0.10)	$0.21$ ( $\pm$ 0.05)	$0.21$ ( $\pm$ 0.07)	$0.19$ ( $\pm$ 0.09)
NoGAM	$0.23$ ( $\pm$ 0.02) / $0.26$ ( $\pm$ 0.01) / $0.29$ ( $\pm$ 0.04)	$0.26$ ( $\pm$ 0.05)	$0.27$ ( $\pm$ 0.04)	$0.24$ ( $\pm$ 0.04)
CAM	$\mathbf{0.41}$ ( $\pm$ 0.06) / $0.42$ ( $\pm$ 0.08) / $0.40$ ( $\pm$ 0.08)	$0.31$ ( $\pm$ 0.09)	$0.34$ ( $\pm$ 0.08)	$0.31$ ( $\pm$ 0.09)
PC	$0.27$ ( $\pm$ 0.10) / $0.26$ ( $\pm$ 0.10) / $0.25$ ( $\pm$ 0.09)	$0.29$ ( $\pm$ 0.08)	$0.27$ ( $\pm$ 0.07)	$0.21$ ( $\pm$ 0.12)
FCI	$0.16$ ( $\pm$ 0.05) / $0.16$ ( $\pm$ 0.06) / $0.13$ ( $\pm$ 0.04)	$0.13$ ( $\pm$ 0.11)	$0.18$ ( $\pm$ 0.07)	$0.16$ ( $\pm$ 0.11)

Appendix

Table 20: F1 (Arrow), two boxes selection

Method	$\rho=1$ $n=1493$ / $747$ / $373$	$\rho=0.2$ ( $n=1493$ )	$\rho=0.1$ ( $n=747$ )	$\rho=0.05$ ( $n=373$ )
SCORE	$0.38$ ( $\pm$ 0.06) / $0.42$ ( $\pm$ 0.07) / $0.42$ ( $\pm$ 0.04)	$0.52$ ( $\pm$ 0.08)	$0.52$ ( $\pm$ 0.08)	$0.52$ ( $\pm$ 0.09)
DAS	$0.38$ ( $\pm$ 0.06) / $0.42$ ( $\pm$ 0.07) / $0.42$ ( $\pm$ 0.04)	$0.52$ ( $\pm$ 0.08)	$0.52$ ( $\pm$ 0.08)	$0.52$ ( $\pm$ 0.09)
NoGAM	$0.42$ ( $\pm$ 0.03) / $0.42$ ( $\pm$ 0.03) / $0.44$ ( $\pm$ 0.04)	$0.53$ ( $\pm$ 0.06)	$0.52$ ( $\pm$ 0.07)	$0.52$ ( $\pm$ 0.06)
CAM	$0.64$ ( $\pm$ 0.07) / $0.61$ ( $\pm$ 0.12) / $0.59$ ( $\pm$ 0.12)	$0.40$ ( $\pm$ 0.12)	$0.45$ ( $\pm$ 0.12)	$0.43$ ( $\pm$ 0.14)
PC	- / - / -	-	-	-
FCI	- / - / -	-	-	-

Appendix

Table 21: Causal ordering accuracy

1-D_{top}/|E|

(0.5 = random order), halfspace selection

Method	$\rho=1$ $n=1493$ / $747$ / $373$	$\rho=0.2$ ( $n=1493$ )	$\rho=0.1$ ( $n=747$ )	$\rho=0.05$ ( $n=373$ )
SCORE	$0.38$ ( $\pm$ 0.06) / $0.42$ ( $\pm$ 0.07) / $0.42$ ( $\pm$ 0.04)	$0.45$ ( $\pm$ 0.07)	$0.48$ ( $\pm$ 0.07)	$0.49$ ( $\pm$ 0.06)
DAS	$0.38$ ( $\pm$ 0.06) / $0.42$ ( $\pm$ 0.07) / $0.42$ ( $\pm$ 0.04)	$0.45$ ( $\pm$ 0.07)	$0.48$ ( $\pm$ 0.07)	$0.49$ ( $\pm$ 0.06)
NoGAM	$0.42$ ( $\pm$ 0.03) / $0.42$ ( $\pm$ 0.03) / $0.44$ ( $\pm$ 0.04)	$0.46$ ( $\pm$ 0.04)	$0.46$ ( $\pm$ 0.05)	$0.47$ ( $\pm$ 0.08)
CAM	$0.64$ ( $\pm$ 0.07) / $0.61$ ( $\pm$ 0.12) / $0.59$ ( $\pm$ 0.12)	$0.49$ ( $\pm$ 0.14)	$0.51$ ( $\pm$ 0.17)	$0.45$ ( $\pm$ 0.19)
PC	- / - / -	-	-	-
FCI	- / - / -	-	-	-

Appendix

Table 22: Causal ordering accuracy

1-D_{top}/|E|

(0.5 = random order), annulus selection

Method	$\rho=1$ $n=1493$ / $747$ / $373$	$\rho=0.2$ ( $n=1493$ )	$\rho=0.1$ ( $n=747$ )	$\rho=0.05$ ( $n=373$ )
SCORE	$0.38$ ( $\pm$ 0.06) / $0.42$ ( $\pm$ 0.07) / $0.42$ ( $\pm$ 0.04)	$0.39$ ( $\pm$ 0.06)	$0.40$ ( $\pm$ 0.07)	$0.40$ ( $\pm$ 0.07)
DAS	$0.38$ ( $\pm$ 0.06) / $0.42$ ( $\pm$ 0.07) / $0.42$ ( $\pm$ 0.04)	$0.39$ ( $\pm$ 0.06)	$0.40$ ( $\pm$ 0.07)	$0.40$ ( $\pm$ 0.07)
NoGAM	$0.42$ ( $\pm$ 0.03) / $0.42$ ( $\pm$ 0.03) / $0.44$ ( $\pm$ 0.04)	$0.45$ ( $\pm$ 0.05)	$0.43$ ( $\pm$ 0.06)	$0.41$ ( $\pm$ 0.06)
CAM	$0.64$ ( $\pm$ 0.07) / $0.61$ ( $\pm$ 0.12) / $0.59$ ( $\pm$ 0.12)	$0.51$ ( $\pm$ 0.14)	$0.52$ ( $\pm$ 0.15)	$0.51$ ( $\pm$ 0.13)
PC	- / - / -	-	-	-
FCI	- / - / -	-	-	-

Appendix

Table 23: Causal ordering accuracy

1-D_{top}/|E|

(0.5 = random order), two boxes selection

Method	$\rho=1.0$	$\rho=0.5$
Method	non-select	halfspace	annulus	two boxes
SCORE	$\mathbf{0.37}$ ( $\pm$ 0.09)	$\mathbf{0.36}$ ( $\pm$ 0.07)	$0.37$ ( $\pm$ 0.08)	$0.38$ ( $\pm$ 0.09)
DAS	$0.31$ ( $\pm$ 0.08)	$0.31$ ( $\pm$ 0.07)	$0.31$ ( $\pm$ 0.08)	$0.32$ ( $\pm$ 0.07)
NoGAM	$0.35$ ( $\pm$ 0.07)	$0.35$ ( $\pm$ 0.07)	$0.35$ ( $\pm$ 0.07)	$0.34$ ( $\pm$ 0.07)
CAM	$0.28$ ( $\pm$ 0.11)	$0.28$ ( $\pm$ 0.09)	$0.28$ ( $\pm$ 0.08)	$0.29$ ( $\pm$ 0.11)
PC	$0.08$ ( $\pm$ 0.05)	$0.16$ ( $\pm$ 0.02)	$0.11$ ( $\pm$ 0.03)	$0.15$ ( $\pm$ 0.03)

Appendix

Table 24: F1 arrow of per-condition Sachs dataset

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