Unified Sharp Bounds on Probabilities of Causation

Xin Shu; Ang Li

cs.AIOct 19, 2026

Unified Sharp Bounds on Probabilities of Causation

Authors: Xin Shu, Ang Li

Organizations: Department of Computer Science, Florida State University

Abstract

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

Panel of Figure 1 — Figure 1: Lemma 12 in one column of mass $c_{\ell}$ , with $t=3$ . We place the complements $N_{i}$ one after another on a circle of circumference $c_{\ell}$ . 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.

Panel of Figure 2 — Figure 2: The network of Lemma 13 for $k=3$ , $n=4$ . Each $A_{j}$ must place mass $d_{j}=\mathbb{P}(A_{j}\setminus x_{j})$ in the columns $x_{\ell}$ , $\ell\neq j$ , at most $c_{\ell}$ in each. There is no edge from $A_{j}$ to $x_{j}$ , because inside its own column $A_{j}$ is fixed to the cell of mass $\beta_{j}$ . Column $x_{\ell}$ can absorb up to $f_{\ell}$ for free, and any inflow beyond that adds to $\theta$ .

Causal Inference

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

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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

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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

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