Probabilities of causation (PoCs) have developed steadily, from the binary measures first introduced by Pearl to the non-binary setting recently studied by Shu et al. As the theory has matured, PoCs have found applications in a growing range of areas, including personalized medicine, marketing and legal attribution. Shu et al. proposed closed-form bounds for multi-valued treatments and outcomes and proved them valid, but left their sharpness in high dimensions open. We close this gap with a mathematical proof. We further show that \PNS, \PSub, \PRep and \PN are all special cases of a single, more general formula. Combined with their equivalence classes, this settles the question of sharp bounds for discrete PoCs from experimental and observational data: in every dimension, they are given in closed form by a single theorem.
Figures & tables
Figure 1: Lemma 12 in one column of mass cℓ , with t=3 . We place the complements Ni one after another on a circle of circumference cℓ . If they fit, they do not overlap and the union bound holds with equality. If they do not fit, they wrap around and cover the whole column, so the intersection is empty.
Figure 2: The network of Lemma 13 for k=3 , n=4 . Each Aj must place mass dj=P(Aj∖xj) in the columns xℓ , ℓ=j , at most cℓ in each. There is no edge from Aj to xj , because inside its own column Aj is fixed to the cell of mass βj . Column xℓ can absorb up to fℓ for free, and any inflow beyond that adds to θ .
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
Probabilities of causation (PoCs), such as the probability of necessity and sufficiency (PNS), are important tools for decision making but are generally not point identifiable. Existing work has derived bounds for these quantities using combinations of experimental and observational data. However, there is very limited research on sample size analysis, namely, how many experimental and observational samples are required to achieve a desired margin of error. In this paper, we propose a general sample size framework based on the delta method. Our approach applies to settings in which the target bounds of PoCs can be expressed as finite minima or maxima of linear combinations of experimental and observational probabilities. Through simulation studies, we demonstrate that the proposed sample size calculations lead to stable estimation of these bounds.
Tianyuan Cheng, Ruirui Mao, Judea Pearl +1
Department of Statistics, Florida State University · Department of Computer Science, Cornell University · Department of Computer Science, University of California Los Angeles +1
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.