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
Figure 1: Causal structures used in the experiments.
n
↑ TP lb
↓ TP ub
↑ MLP lb
↓ 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) .
↑ TP lb
↓ TP ub
↑ MLP lb
↓ 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) .
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
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
This paper investigates the development of causal foundation models for bounding the effect of interventions and counterfactuals from observational data. We show that a canonical prior can be defined with full support over the space of structural causal models with discrete observables. With this canonical prior, we translate the problem of bounding counterfactuals into that of learning distributions over functions that map data (and possibly structural assumptions) to a causal query of interest. This extends the promising causal foundational modelling paradigm to the estimation of partially-identifiable causal effects, i.e., under unobserved confounding, where multiple values are equally compatible with the observed data and prior structural assumptions.