  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.
