Probabilities of Causation for Continuous and Vector Variables

TL;DR

This work extends probabilities of causation (PoC) to continuous and vector-valued treatments and outcomes, introducing nonparametric identification theorems under exogeneity and monotonicity-like assumptions. It develops scalar PoC definitions and their identifiability, then generalizes to multivariate outcomes using a lexicographic order and conditional CDFs, enabling PoC analysis with covariates. The authors further propose PoC variants with evidence and multi-hypothetical terms, provide corresponding identification results, and demonstrate the approach on a real-world education dataset. Overall, the framework broadens causal reasoning with continuous and high-dimensional effects, offering practical tools for decision-making and explainable AI.

Abstract

Probabilities of causation (PoC) are valuable concepts for explainable artificial intelligence and practical decision-making. PoC are originally defined for scalar binary variables. In this paper, we extend the concept of PoC to continuous treatment and outcome variables, and further generalize PoC to capture causal effects between multiple treatments and multiple outcomes. In addition, we consider PoC for a sub-population and PoC with multi-hypothetical terms to capture more sophisticated counterfactual information useful for decision-making. We provide a nonparametric identification theorem for each type of PoC we introduce. Finally, we illustrate the application of our results on a real-world dataset about education.

Figures (3)

• Figure 1: Trajectories for (1) $Y_x(u_{\rho(y;x_0)})$ and (2) $Y_x(u_{\rho(y;x_1)})$.
• Figure 2: Trajectories for (1) ${\boldsymbol Y}_{\boldsymbol x}({\boldsymbol u}_{\rho({\boldsymbol y};{\boldsymbol x}_0,{\boldsymbol c})})$, (2) ${\boldsymbol Y}_{\boldsymbol x}({\boldsymbol u}_{\rho({\boldsymbol y};{\boldsymbol x}_1,{\boldsymbol c})})$, (3) ${\boldsymbol Y}_{\boldsymbol x}({\boldsymbol u}_{\rho^{o}({\boldsymbol y}';{\boldsymbol x}',{\boldsymbol c})})$ and (4) ${\boldsymbol Y}_{\boldsymbol x}({\boldsymbol u}_{\rho({\boldsymbol y}';{\boldsymbol x}',{\boldsymbol c})})$.
• Figure 3: Trajectories for (1) ${\boldsymbol Y}_{\boldsymbol x}({\boldsymbol u}_{\rho({\boldsymbol y}_1;{\boldsymbol x}_0,{\boldsymbol c})})$, (2) ${\boldsymbol Y}_{\boldsymbol x}({\boldsymbol u}_{\rho({\boldsymbol y}_1;{\boldsymbol x}_1,{\boldsymbol c})})$, (3) ${\boldsymbol Y}_{\boldsymbol x}({\boldsymbol u}_{\rho({\boldsymbol y}_2;{\boldsymbol x}_1,{\boldsymbol c})})$ and (4) ${\boldsymbol Y}_{\boldsymbol x}({\boldsymbol u}_{\rho({\boldsymbol y}_2;{\boldsymbol x}_2,{\boldsymbol c})})$.

Theorems & Definitions (31)

• Definition 2.1: PoC
• Definition 3.1: Probabilities of causation
• Lemma 3.1
• Example 3.1
• Theorem 3.1
• Theorem 3.2: Identification of PoC
• Definition 4.1: Conditional PoC
• Theorem 4.1
• Theorem 4.2: Identification of conditional PoC
• Definition 5.1: Conditional PNS with evidence $({\boldsymbol y}',{\boldsymbol x}',{\boldsymbol c})$