Bounding Probabilities of Causation with Partial Causal Diagrams

Yuxuan Xie; Ang Li

cs.AIFeb 16, 2026

Bounding Probabilities of Causation with Partial Causal Diagrams

Authors: Yuxuan Xie, Ang Li

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

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1: Causal structures used in the experiments.

$n$	$\uparrow$ TP lb	$\downarrow$ TP ub	$\uparrow$ MLP lb	$\downarrow$ MLP ub	gap (TP)	gap (MLP)	gap (Col. 3.2 )	# imp. TP	# imp. MLP
1	0.0263	0.0251	0	0	0.2136	0.1622	0.1622	634	0
2	0.0407	0.0421	0.0007	0.0009	0.2690	0.1878	0.1862	848	565
3	0.0453	0.0447	0.0020	0.0020	0.3101	0.2242	0.2201	927	673
4	0.0478	0.0478	0.0033	0.0033	0.3469	0.2578	0.2512	965	727
5	0.0436	0.0422	0.0037	0.0035	0.3693	0.2907	0.2835	991	787
6	0.0402	0.0398	0.0043	0.0039	0.3965	0.3247	0.3165	996	818

Table 1: Performance comparison of bounds with multiple non-descendant covariates using covariate-specific data.

Figure 2: PNS lower bounds for $n=6$ in Figure 1(a) .

Figure 3: PNS lower bounds for $n=6$ in Figure 1(a) .

Figure 4: PNS lower bounds for $n=5$ in Figure 1(a) .

Figure 5: PNS lower bounds for $n=5$ in Figure 1(a) .

Figure 6: PNS lower bounds for $n=4$ in Figure 1(a) .

Figure 7: PNS lower bounds for $n=4$ in Figure 1(a) .

$\uparrow$ TP lb	$\downarrow$ TP ub	$\uparrow$ MLP lb	$\downarrow$ MLP ub	gap (TP)	gap (MLP)	gap (Thm 3.3 )	# impr. TP	# impr. MLP
0.0268	0.0258	0.0057	0.0051	0.2833	0.2417	0.2308	782	245

Table 2: Performance comparison for the mediator case with a back-door set under the causal structure in Figure 1(b) .

Figure 8: PNS lower bounds comparison in Figure 1(b) .

Figure 9: PNS upper bounds comparison in Figure 1(b) .

Causal Inference

Causal inference studies how interventions and counterfactuals reveal cause-and-effect relationships beyond observed associations.

See all 307 papers

General Probabilities of Causation with Causal Knowledge
Aug 12, 2026Xin Shu, Zhen Lei, Ang LiPartial Identification Causal Identifiability
Department of Computer Science, Florida State University
Partial Identification under Causal Orders by Linear Programming
Aug 25, 2026Eric Rossetto, Alessandro AntonucciPartial Identification Causal Inference
Istituto Dalle Molle di Studi sull’Intelligenza Artificiale (IDSIA) Scuola Universitaria Professionale della Svizzera Italiana (SUPSI) Lugano, Switzerland
Foundation Models for Partial Causal Identification
Aug 21, 2026Alexis Bellot, Anish DhirCausal Foundation Models Causal Effect Estimation
Independent · University College London

Aug 12, 2026J/K move · Enter open · S save

General Probabilities of Causation with Causal Knowledge

Xin Shu, Zhen Lei, Ang Li

Department of Computer Science, Florida State University

Partial Identification Causal Identifiability Causal Inference

Open paper AskPDF arXiv

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.

Abstract

Causal Inference

Explore similar work

General Probabilities of Causation with Causal Knowledge

Partial Identification under Causal Orders by Linear Programming

Foundation Models for Partial Causal Identification