Bounds and Identification of Joint Probabilities of Potential Outcomes and Observed Variables under Monotonicity Assumptions

Naoya Hashimoto; Yuta Kawakami; Jin Tian

stat.MLFeb 21, 2026

Bounds and Identification of Joint Probabilities of Potential Outcomes and Observed Variables under Monotonicity Assumptions

Authors: Naoya Hashimoto, Yuta Kawakami, Jin Tian

Organizations: Mohamed bin Zayed University of Artificial Intelligence

Abstract

Evaluating joint probabilities of potential outcomes and observed variables, and their linear combinations, is a fundamental challenge in causal inference. This paper addresses the bounding and identification of these probabilities in settings with discrete treatment and discrete ordinal outcome. We propose new families of monotonicity assumptions and formulate the bounding problem as a linear programming problem. We further introduce a new monotonicity assumption specifically to achieve identification. Finally, we present numerical experiments to validate our methods and demonstrate their application using real-world datasets.

Figures & tables

Research	Treatment Type	Outcome Type	MA	Objective	Identification
Pearl (1999) ; Tian and Pearl (2000)	Binary	Binary	$\mathbb{P}(Y_{0}=1,Y_{1}=0)=0$	PoC	$\checkmark$
Mueller and Pearl (2023)	Binary	Binary	$0\leq\mathbb{P}(Y_{0}=1,Y_{1}=0)\leq\epsilon$	PoC
Zhang et al. (2025)	Binary	Ordinal	$Y_{0}\leq Y_{1}$	PN
Zhang et al. (2025)	Binary	Ordinal	$0\leq Y_{1}-Y_{0}\leq 1$	PN	$\checkmark$
Manski (1997)	Discrete	General	$Y_{0}\leq Y_{1}\leq\dots\leq Y_{d_{X}-1}$	ACE
This paper	Discrete	Ordinal	Assumptions 7 $\sim$ 11	$\lambda({\cal P})$

Table 1: Summary of monotonicity assumptions (MAs) in causal inference. The symbol (

\checkmark

) indicates that the objective quantity is identified under the MA.

\lambda({\cal P})

represents an arbitrary linear combination of the joint POs/OVs probabilities.

Assumption	LB	UB
(A1)	0.000 [0.000,0.000]	0.261 [0.247,0.274]
(A2)	0.064 [0.042,0.089]	0.261 [0.247,0.274]
(A3)	0.000 [0.000,0.000]	0.064 [0.042,0.089]
(A4)	0.014 [0.000,0.039]	0.114 [0.092,0.139]
Assumption	Identification
(A5)	0.064 [0.042,0.089]

Table 2: The means and 95% CIs of the estimates of

\mathbb{P}(Y_{0}=0,Y_{1}=0,Y_{2}\geq 1)

. LB stands for “lower bound” and UB stands for “upper bound.”

Assumption	LB	UB
(B1)	0.203 [0.000,0.489]	0.997 [0.956,1.000]
(B2)	0.293 [0.000,0.561]	0.997 [0.956,1.000]
(B3)	0.203 [0.000,0.489]	0.575 [0.192,0.740]
(B4)	0.207 [0.000,0.489]	0.783 [0.531,1.000]
Assumption	Identification
(B5)	0.575 [0.192,0.740]

Table 3: The means and 95% CIs of the estimates of

\mathbb{P}(Y_{0}\leq 1,Y_{1}\leq 1,Y_{2}\leq 1|X=3,Y=2)

under exogeneity.

Assumption	LB	UB
(B1)	0.044 [0.000,0.280]	1.000 [1.000,1.000]
(B2)	0.044 [0.000,0.280]	0.251 [0.013,0.612]
(B3)	0.044 [0.000,0.280]	1.000 [1.000,1.000]
(B4)	0.044 [0.000,0.280]	0.404 [0.165,0.665]
Assumption	Identification
(B5)	0.251 [0.013,0.612]

Table 4: The means and 95% CIs of the estimates of

\mathbb{P}(Y_{1}\geq 1,Y_{2}\geq 1,Y_{3}\geq 1|X=0,Y=0)

under exogeneity.

Appendix figures & tables24 assets

Supplementary material from the paper’s appendix.

Appendix

$X=0,Y=0$	$Y_{0}=0$			$Y_{0}=1$			$Y_{0}=2$
	$Y_{1}=0$	$Y_{1}=1$	$Y_{1}=2$	$Y_{1}=0$	$Y_{1}=1$	$Y_{1}=2$	$Y_{1}=0$	$Y_{1}=1$	$Y_{1}=2$
$Y_{2}=0$	1/40	0	0	0	0	0	0	0	0
$Y_{2}=1$	1/40	1/40	0	0	0	0	0	0	0
$Y_{2}=2$	1/40	1/40	1/40	0	0	0	0	0	0

Appendix

Table 5: Setting of simulations for Assumptions (i)-(v) in Section E.1

$X=0,Y=0$	$Y_{0}=0$			$Y_{0}=1$			$Y_{0}=2$
	$Y_{1}=0$	$Y_{1}=1$	$Y_{1}=2$	$Y_{1}=0$	$Y_{1}=1$	$Y_{1}=2$	$Y_{1}=0$	$Y_{1}=1$	$Y_{1}=2$
$Y_{2}=0$	0.011	0.010	0.042	0	0	0	0	0	0
$Y_{2}=1$	0.045	0.059	0.069	0	0	0	0	0	0
$Y_{2}=2$	0.017	0.041	0.047	0	0	0	0	0	0

Appendix

Table 6: Settings of

\mathbb{P}(Y_{1},Y_{0},Y_{2},X,Y)

for comparing bounds based on [ Li and Pearl, 2024 ] and LP solver. When

x\neq y

, we set

\mathbb{P}(Y_{0},Y_{1},Y_{2},X=x,Y=y)=0

$X=0,Y=0$	$Y_{0}=0$			$Y_{0}=1$			$Y_{0}=2$
	$Y_{1}=0$	$Y_{1}=1$	$Y_{1}=2$	$Y_{1}=0$	$Y_{1}=1$	$Y_{1}=2$	$Y_{1}=0$	$Y_{1}=1$	$Y_{1}=2$
$Y_{2}=0$	1/20	0	0	0	0	0	0	0	0
$Y_{2}=1$	1/20	1/20	0	0	0	0	0	0	0
$Y_{2}=2$	0	0	0	0	0	0	0	0	0

Appendix

Table 7: Setting of simulations for Assumption (vi) in Section E.1

Exp.
MAs	$N=10$	$N=100$	$N=1000$	$N=10000$
(i)	3	3	3	3
(ii)	3	3	3	3
(iii)	3	3	3	3
(iv)	3	4	3	3

Appendix

Table 8: Running time of solving maximization and minimization problem of

\mathbb{P}(Y_{0}=0,Y_{1}=0,Y_{2}=1)

for one time in milliseconds. Each entry is the mean over 100 repetitions.

Exp.
MAs	$N=10$	$N=100$	$N=1000$	$N=10000$
(i)	2	3	2	3
(ii)	3	3	3	3
(iii)	3	3	2	2
(iv)	3	2	2	2

Appendix

Table 9: Running time of solving maximization and minimization problem of

\mathbb{E}[Y_{1}-Y_{0}|X=2,Y=2]

for one time in milliseconds. Each entry is the mean over 100 repetitions.

MAs	Bounds (Exp.)	$N=10$	$N=100$	$N=1000$	$N=10000$
(i)	LB	0.000 [0.000,0.000]	0.000 [0.000,0.000]	0.000 [0.000,0.000]	0.000 [0.000,0.000]
(i)	UB	0.224 [0.100,0.452]	0.265 [0.200,0.340]	0.274 [0.245,0.304]	0.275 [0.268,0.284]
(ii)	LB	0.002 [0.000,0.000]	0.000 [0.000,0.000]	0.000 [0.000,0.000]	0.000 [0.000,0.000]
(ii)	UB	0.224 [0.100,0.452]	0.265 [0.200,0.340]	0.274 [0.245,0.304]	0.275 [0.268,0.284]
(iii)	LB	0.000 [0.000,0.000]	0.000 [0.000,0.000]	0.000 [0.000,0.000]	0.000 [0.000,0.000]
(iii)	UB	0.163 [0.000,0.400]	0.167 [0.105,0.235]	0.166 [0.144,0.186]	0.167 [0.161,0.175]

Appendix

Table 10: Results of the estimates of the bounds of

\mathbb{P}(Y_{0}=0,Y_{1}=0,Y_{2}=1)

. We report the means and 95% confidence intervals (CIs). The true value of

\mathbb{P}(Y_{0}=0,Y_{1}=0,Y_{2}=1)

is 0.092.

MAs	Bounds (Obs.)	$N=10$	$N=100$	$N=1000$	$N=10000$
(i)	LB	-2.000 [-2.000,-2.000]	-2.000 [-2.000,-2.000]	-2.000 [-2.000,-2.000]	-2.000 [-2.000,-2.000]
(i)	UB	2.000 [2.000,2.000]	2.000 [2.000,2.000]	2.000 [2.000,2.000]	2.000 [2.000,2.000]
(ii)	LB	0.000 [0.000,0.000]	0.000 [0.000,0.000]	0.000 [0.000,0.000]	0.000 [0.000,0.000]
(ii)	UB	2.000 [2.000,2.000]	2.000 [2.000,2.000]	2.000 [2.000,2.000]	2.000 [2.000,2.000]
(iii)	LB	-2.000 [-2.000,-2.000]	-2.000 [-2.000,-2.000]	-2.000 [-2.000,-2.000]	-2.000 [-2.000,-2.000]
(iii)	UB	2.000 [2.000,2.000]	2.000 [2.000,2.000]	2.000 [2.000,2.000]	2.000 [2.000,2.000]

Appendix

Table 11: Results of the estimates of the bounds of

\mathbb{E}[Y_{1}-Y_{0}|X=2,Y=2]

. We report the means and 95% confidence intervals (CIs). The true value of

\mathbb{E}[Y_{1}-Y_{0}|X=2,Y=2]

is 0.667.

MAs	Bounds (Exp.)	$N=10$	$N=100$	$N=1000$	$N=10000$
(i)	LB	0.417 [0.047,0.952]	0.441 [0.325,0.570]	0.432 [0.393,0.472]	0.432 [0.422,0.443]
(i)	UB	2.003 [0.900,3.052]	2.176 [1.803,2.485]	2.189 [2.077,2.313]	2.197 [2.151,2.240]
(ii)	LB	0.417 [0.047,0.952]	0.441 [0.325,0.570]	0.432 [0.393,0.472]	0.432 [0.422,0.443]
(ii)	UB	0.629 [0.048,1.300]	0.764 [0.539,1.016]	0.745 [0.659,0.816]	0.747 [0.725,0.771]
(iii)	LB	0.417 [0.047,0.952]	0.441 [0.325,0.570]	0.432 [0.393,0.472]	0.432 [0.422,0.443]
(iii)	UB	2.003 [0.900,3.052]	2.176 [1.803,2.485]	2.189 [2.077,2.313]	2.197 [2.151,2.240]

Appendix

Table 12: Results of the estimates of the bounds of

\mathbb{E}[(Y_{1}-Y_{0})^{2}]

. We report the means and 95% confidence intervals (CIs). The true value of

\mathbb{E}[(Y_{1}-Y_{0})^{2}]

is 0.583.

Bounds (Both)	$N=10$	$N=100$	$N=1000$	$N=10000$
LB ( [ Li and Pearl, 2024 ] )	0.056 [0.000,0.300]	0.003 [0.000,0.025]	0.000 [0.000,0.000]	0.000 [0.000,0.000]
UB ( [ Li and Pearl, 2024 ] )	0.426 [0.100,0.700]	0.501 [0.410,0.570]	0.517 [0.485,0.549]	0.514 [0.505,0.526]
LB	0.056 [0.000,0.300]	0.003 [0.000,0.025]	0.000 [0.000,0.000]	0.000 [0.000,0.000]
UB	0.389 [0.100,0.600]	0.414 [0.305,0.508]	0.426 [0.387,0.451]	0.426 [0.411,0.435]

Appendix

Table 13: The bound of

\mathbb{P}(Y_{0}=0,Y_{1}=1,Y_{2}=2)

based on Theorem 11 in [ Li and Pearl, 2024 ] and LP solver under the setting in Table 6 . The true value of

\mathbb{P}(Y_{0}=0,Y_{1}=1,Y_{2}=2)

is 0.214.

Data	$L$	LB	UB
Obs	(All)	0.000 [0.000, 0.000]	0.351 [0.326, 0.374]
Exp. & Both	0.00	0.000 [0.000, 0.000]	0.274 [0.250, 0.301]
Exp. & Both	0.70	0.000 [0.000, 0.000]	0.274 [0.250, 0.301]
Exp. & Both	0.75	0.000 [0.000, 0.000]	0.274 [0.250, 0.301]
Exp. & Both	0.80	0.000 [0.000, 0.000]	0.274 [0.250, 0.301]
Exp. & Both	0.85	0.000 [0.000, 0.000]	0.274 [0.250, 0.301]

Appendix

Table 14: Results of the estimates of the bounds of

\mathbb{P}(Y_{0}=0,Y_{1}=0,Y_{2}=1)

under assumption

L\leq\mathbb{P}(Y_{0}\leq Y_{1}\leq Y_{2})

and

N=1000

. We report the means and 95% confidence intervals (CIs). The true value of

\mathbb{P}(Y_{0}=0,Y_{1}=0,Y_{2}=1)

is 0.092.

Panel of Figure 1 — Figure 1: Results of the estimates of the bounds of $\mathbb{P}(Y_{0}=0,Y_{1}=0,Y_{2}=1)$ as $L$ varies. The x-axis shows the value of $L$ , and the y-axis shows the mean estimates of the bounds of $\mathbb{P}(Y_{0}=0,Y_{1}=0,Y_{2}=1)$ .

Data	$L$	LB	UB
Obs	(All)	-2.000 [-2.000, -2.000]	2.000 [2.000, 2.000]
Both	0.00	-1.509 [-1.850, -1.267]	2.000 [2.000, 2.000]
Both	0.70	-1.509 [-1.850, -1.267]	2.000 [2.000, 2.000]
Both	0.75	-1.509 [-1.850, -1.267]	2.000 [2.000, 2.000]
Both	0.80	-1.509 [-1.850, -1.267]	2.000 [2.000, 2.000]
Both	0.85	-1.509 [-1.850, -1.267]	2.000 [2.000, 2.000]

Appendix

Table 15: Results of the estimates of the bounds of

\mathbb{E}[Y_{1}-Y_{0}|X=2,Y=2]

under assumption

L\leq\mathbb{P}(Y_{0}\leq Y_{1}\leq Y_{2})

and

N=1000

. We report the means and 95% confidence intervals (CIs). The true value of

\mathbb{E}[Y_{1}-Y_{0}|X=2,Y=2]

is 0.667.

Panel of Figure 2 — Figure 2: Results of the estimates of the bounds of $\mathbb{E}[Y_{1}-Y_{0}|X=2,Y=2]$ as $L$ varies. The x-axis shows the value of $L$ , and the y-axis shows the mean estimates of the bounds of $\mathbb{E}[Y_{1}-Y_{0}|X=2,Y=2]$ .

Data	$L$	LB	UB
Exp.	0.00	0.435 [0.392, 0.474]	2.196 [2.073, 2.339]
Exp.	0.70	0.435 [0.392, 0.474]	2.195 [2.073, 2.331]
Exp.	0.75	0.435 [0.392, 0.474]	2.141 [2.003, 2.272]
Exp.	0.80	0.435 [0.392, 0.474]	2.041 [1.903, 2.172]
Exp.	0.85	0.435 [0.392, 0.474]	1.921 [1.803, 2.024]
Exp.	0.86	0.435 [0.392, 0.474]	1.863 [1.775, 1.952]

Appendix

Table 16: Results of the estimates of the bounds of

\mathbb{E}[(Y_{1}-Y_{0})^{2}]

under assumption

L\leq\mathbb{P}(Y_{0}\leq Y_{1}\leq Y_{2})

and

N=1000

. We report the means and 95% confidence intervals (CIs). The true value of

\mathbb{E}[(Y_{1}-Y_{0})^{2}]

is 0.583.

Panel of Figure 3 — Figure 3: Results of the estimates of the bounds of $\mathbb{E}[(Y_{1}-Y_{0})^{2}]$ as $L$ varies. The x-axis shows the value of $L$ , and the y-axis shows the mean estimates of the bounds of $\mathbb{E}[(Y_{1}-Y_{0})^{2}]$ .

$L_{3}=0.900$
		$L_{2}=0.900$	$L_{2}=0.925$	$L_{2}=0.950$	$L_{2}=0.975$	$L_{2}=1.000$
$L_{1}=0.900$	LB	0.000 [0.000, 0.000]	0.000 [0.000, 0.000]	0.001 [0.000, 0.008]	0.009 [0.000, 0.033]	0.033 [0.012, 0.058]
$L_{1}=0.900$	UB	0.233 [0.212, 0.258]	0.208 [0.187, 0.233]	0.183 [0.162, 0.208]	0.158 [0.137, 0.183]	0.133 [0.112, 0.158]
$L_{1}=0.925$	LB	0.000 [0.000, 0.000]	0.001 [0.000, 0.008]	0.009 [0.000, 0.033]	0.033 [0.012, 0.058]	0.058 [0.037, 0.083]
$L_{1}=0.925$	UB	0.233 [0.212, 0.258]	0.208 [0.187, 0.233]	0.183 [0.162, 0.208]	0.158 [0.137, 0.183]	0.133 [0.112, 0.158]
$L_{1}=0.950$	LB	0.001 [0.000, 0.008]	0.009 [0.000, 0.033]	0.033 [0.012, 0.058]	0.058 [0.037, 0.083]	0.083 [0.062, 0.108]

Appendix

Table 17: Results of the estimates of bounds of

\mathbb{P}(Y_{0}=0,Y_{1}=0,Y_{2}=1)

under Assumption (vi) and

N=1000

with experimental data / with both experimental and observational data. The true value of

\mathbb{P}(Y_{0}=0,Y_{1}=0,Y_{2}=1)

is 0.133.

$L_{3}=0.900$
		$L_{2}=0.900$	$L_{2}=0.925$	$L_{2}=0.950$	$L_{2}=0.975$	$L_{2}=1.000$
$L_{1}=0.900$	LB	-1.929 [-2.000, -1.667]	-1.713 [-1.909, -1.458]	-1.464 [-1.606, -1.250]	-1.214 [-1.303, -1.042]	-0.965 [-1.000, -0.833]
$L_{1}=0.900$	UB	1.965 [1.833, 2.000]	1.965 [1.833, 2.000]	1.965 [1.833, 2.000]	1.965 [1.833, 2.000]	1.965 [1.833, 2.000]
$L_{1}=0.925$	LB	-1.498 [-1.819, -1.250]	-1.498 [-1.819, -1.250]	-1.248 [-1.516, -1.042]	-0.998 [-1.213, -0.833]	-0.749 [-0.909, -0.625]
$L_{1}=0.925$	UB	1.749 [1.625, 1.909]	1.749 [1.625, 1.909]	1.749 [1.625, 1.909]	1.749 [1.625, 1.909]	1.749 [1.625, 1.909]
$L_{1}=0.950$	LB	-0.998 [-1.213, -0.833]	-0.998 [-1.213, -0.833]	-0.998 [-1.213, -0.833]	-0.749 [-0.909, -0.625]	-0.499 [-0.606, -0.417]

Appendix

Table 18: Results of the estimates of bounds of

\mathbb{E}[Y_{1}-Y_{0}|X=2,Y=2]

under Assumption (vi) and

N=1000

with observational data. The true value of

\mathbb{E}[Y_{1}-Y_{0}|X=2,Y=2]

is 0.333.

$L_{3}=0.900$
		$L_{2}=0.900$	$L_{2}=0.925$	$L_{2}=0.950$	$L_{2}=0.975$	$L_{2}=1.000$
$L_{1}=0.900$	LB	-1.758 [-2.000, -1.436]	-1.556 [-1.835, -1.241]	-1.307 [-1.546, -1.022]	-1.057 [-1.268, -0.795]	-0.808 [-1.000, -0.564]
$L_{1}=0.900$	UB	1.926 [1.767, 2.000]	1.926 [1.767, 2.000]	1.926 [1.767, 2.000]	1.926 [1.767, 2.000]	1.917 [1.752, 2.000]
$L_{1}=0.925$	LB	-1.475 [-1.769, -1.248]	-1.461 [-1.769, -1.198]	-1.216 [-1.467, -0.998]	-0.966 [-1.164, -0.788]	-0.716 [-0.862, -0.564]
$L_{1}=0.925$	UB	1.744 [1.625, 1.909]	1.744 [1.625, 1.909]	1.744 [1.625, 1.909]	1.744 [1.625, 1.909]	1.727 [1.551, 1.909]
$L_{1}=0.950$	LB	-0.998 [-1.213, -0.833]	-0.998 [-1.213, -0.833]	-0.998 [-1.213, -0.833]	-0.749 [-0.909, -0.625]	-0.499 [-0.606, -0.417]

Appendix

Table 19: Results of the estimates of bounds of

\mathbb{E}[Y_{1}-Y_{0}|X=2,Y=2]

under Assumption (vi) and

N=1000

with both experimental and observational data. The true value of

\mathbb{E}[Y_{1}-Y_{0}|X=2,Y=2]

is 0.333.

$L_{3}=0.900$
		$L_{2}=0.900$	$L_{2}=0.925$	$L_{2}=0.950$	$L_{2}=0.975$	$L_{2}=1.000$
$L_{1}=0.900$	LB	0.269 [0.242, 0.298]	0.269 [0.242, 0.298]	0.269 [0.242, 0.298]	0.269 [0.242, 0.298]	0.269 [0.242, 0.298]
$L_{1}=0.900$	UB	0.869 [0.842, 0.898]	0.869 [0.842, 0.898]	0.869 [0.842, 0.898]	0.869 [0.842, 0.898]	0.869 [0.842, 0.898]
$L_{1}=0.925$	LB	0.269 [0.242, 0.298]	0.269 [0.242, 0.298]	0.269 [0.242, 0.298]	0.269 [0.242, 0.298]	0.269 [0.242, 0.298]
$L_{1}=0.925$	UB	0.719 [0.692, 0.748]	0.719 [0.692, 0.748]	0.719 [0.692, 0.748]	0.719 [0.692, 0.748]	0.719 [0.692, 0.748]
$L_{1}=0.950$	LB	0.269 [0.242, 0.298]	0.269 [0.242, 0.298]	0.269 [0.242, 0.298]	0.269 [0.242, 0.298]	0.269 [0.242, 0.298]

Appendix

Table 20: Results of the estimates of bounds of

\mathbb{E}[(Y_{1}-Y_{0})^{2}]

under Assumption (vi) and

N=1000

with experimental data / with both experimental and observational data. The true value of

\mathbb{E}[(Y_{1}-Y_{0})^{2}]

is 0.267.

$L_{3}=0.900$
		$L_{2}=0.900$	$L_{2}=0.925$	$L_{2}=0.950$	$L_{2}=0.975$	$L_{2}=1.000$
$L_{1}=0.900$	LB	0.000 [0.000, 0.000]	0.000 [0.000, 0.000]	0.000 [0.000, 0.000]	0.000 [0.000, 0.000]	0.000 [0.000, 0.000]
$L_{1}=0.900$	UB	1.200 [1.175, 1.219]	1.200 [1.175, 1.219]	1.197 [1.172, 1.218]	1.175 [1.149, 1.198]	1.150 [1.124, 1.173]
$L_{1}=0.925$	LB	0.000 [0.000, 0.000]	0.000 [0.000, 0.000]	0.000 [0.000, 0.000]	0.000 [0.000, 0.000]	0.000 [0.000, 0.000]
$L_{1}=0.925$	UB	1.100 [1.075, 1.119]	1.100 [1.075, 1.119]	1.097 [1.072, 1.118]	1.075 [1.049, 1.098]	1.050 [1.024, 1.073]
$L_{1}=0.950$	LB	0.000 [0.000, 0.000]	0.000 [0.000, 0.000]	0.000 [0.000, 0.000]	0.000 [0.000, 0.000]	0.000 [0.000, 0.000]

Appendix

Table 21: Results of the estimates of bounds of

\mathbb{E}[(Y_{1}-Y_{0})^{2}]

under Assumption (vi) and

N=1000

with observational data. The true value of

\mathbb{E}[(Y_{1}-Y_{0})^{2}]

is 0.267.

$X=0,Y=0$	$Y_{0}=0$			$Y_{0}=1$			$Y_{0}=2$
	$Y_{1}=0$	$Y_{1}=1$	$Y_{1}=2$	$Y_{1}=0$	$Y_{1}=1$	$Y_{1}=2$	$Y_{1}=0$	$Y_{1}=1$	$Y_{1}=2$
$Y_{2}=0$	1/21	0	0	0	0	0	0	0	0
$Y_{2}=1$	1/21	1/21	0	0	0	0	0	0	0
$Y_{2}=2$	0	0	0	0	0	0	0	0	0

Appendix

Table 22: Setting of Simulations

MAs	Data	$N=10$	$N=100$	$N=1000$	$N=10000$
(I)	Exp.	0.142 [0.000,0.353]	0.145 [0.085,0.225]	0.142 [0.120,0.166]	0.143 [0.135,0.150]
(II)	Obs.	0.190 [0.000,0.500]	0.143 [0.020,0.309]	0.152 [0.085,0.226]	0.144 [0.128,0.157]

Appendix

Table 23: Results of the estimates of

\mathbb{P}(Y_{0}=0,Y_{1}=0,Y_{2}=1)

under identification assumptions. We report the means and 95% confidence intervals (CIs). The ground truth of

\mathbb{P}(Y_{0}=0,Y_{1}=0,Y_{2}=1)

is 0.143.

MAs	Data	$N=10$	$N=100$	$N=1000$	$N=10000$
(II)	Obs.	0.431 [0.000,1.000]	0.332 [0.042,0.700]	0.321 [0.202,0.465]	0.334 [0.288,0.382]

Appendix

Table 24: Results of the estimates of

\mathbb{E}[Y_{1}-Y_{0}|X=2,Y=2]

under identification assumptions. We report the means and 95% confidence intervals (CIs). The ground truth of

\mathbb{E}[Y_{1}-Y_{0}|X=2,Y=2]

is 0.333.

MAs	Data	$N=10$	$N=100$	$N=1000$	$N=10000$
(I)	Exp.	0.313 [0.000,0.600]	0.279 [0.200,0.350]	0.288 [0.264,0.314]	0.285 [0.278,0.294]
(II)	Obs.	0.395 [0.000,1.000]	0.303 [0.077,0.561]	0.285 [0.169,0.388]	0.282 [0.238,0.319]

Appendix

Table 25: Results of the estimates of

\mathbb{E}[(Y_{1}-Y_{0})^{2}]

under identification assumptions. We report the means and 95% confidence intervals (CIs). The ground truth of

\mathbb{E}[(Y_{1}-Y_{0})^{2}]

is 0.286.

Monotonicity Constraints

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Feb 16, 2026

Bounding Probabilities of Causation with Partial Causal Diagrams

Probabilities of causation are fundamental to individual-level explanation and decision making, yet they are inherently counterfactual and not point-identifiable from data in general. Existing bounds either disregard available covariates, require complete causal graphs, or rely on restrictive binary settings, limiting their practical use. In real-world applications, causal information is often partial but nontrivial. This paper proposes a general framework for bounding probabilities of causation using partial causal information. We show how the available structural or statistical information can be systematically incorporated as constraints in a optimization programming formulation, yielding tighter and formally valid bounds without full identifiability. This approach extends the applicability of probabilities of causation to realistic settings where causal knowledge is incomplete but informative.

Yuxuan Xie, Ang Li

University of Ottawa, Ontario, Canada · Department of Computer Science, Florida State University, FL, U.S.A.

Causal Inference Causal Identifiability

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Aug 25, 2026

Partial Identification under Causal Orders by Linear Programming

Non-parametric (partial) identification of counterfactual queries typically relies on a fully specified causal graph. Motivated by settings with incomplete domain knowledge, we challenge this requirement by leveraging structural assumptions that are inherently implied by the query itself. We show that any counterfactual inquiry induces a, mostly partial, topological ordering over relevant variables, which, in turn, enables an explicit query parametrisation reducing the identification task to a linear program. This allows bounding arbitrary counterfactual and nested counterfactual queries. Our work can be viewed as a generalisation of the classical bounding framework of Tian and Pearl (2000), originally developed for probabilities of causation. We also prove the tightness of our bounds by constructing structural causal models that attain the bounds whilst being compatible with both the observed data and the query-implied order. To assess both the generality and practical utility of the proposed bounding procedure, we revisit several case studies from the literature, demonstrating how the derived bounds can be used to yield informative insights even in the absence of an input causal graph.

Eric Rossetto, Alessandro Antonucci

Istituto Dalle Molle di Studi sull’Intelligenza Artificiale (IDSIA) Scuola Universitaria Professionale della Svizzera Italiana (SUPSI) Lugano, Switzerland

Partial Identification Causal Inference Causal Identifiability Structural Causal Models

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Aug 12, 2026

General Probabilities of Causation with Causal Knowledge

Probabilities of causation (PoCs) characterize individual causal responses that cannot be directly observed and therefore generally require partial identification. Tian and Pearl first derived theoretically sharp bounds for binary PoCs, including the probability of necessity (PN), the probability of sufficiency (PS), and the probability of necessity and sufficiency (PNS). Mueller et al. subsequently tightened the bounds for binary PNS by incorporating causal information encoded in covariates and mediators. More recently, Li and Pearl, as well as Shu et al., extended PoCs to multivalued settings and derived corresponding theoretical bounds. These developments naturally raise the question of whether additional causal knowledge can further tighten the bounds in multivalued settings. This paper addresses this question by deriving tighter bounds for multivalued PoCs through the incorporation of causal information encoded in covariates and mediators. We illustrate the theoretical results with toy examples, while simulation studies further demonstrate that the proposed bounds are tighter than existing nonbinary bounds.

Xin Shu, Zhen Lei, Ang Li

Department of Computer Science, Florida State University

Partial Identification Causal Identifiability Causal Inference

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Abstract

Monotonicity Constraints

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Bounding Probabilities of Causation with Partial Causal Diagrams

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