Probabilities of Causation with Nonbinary Treatment and Effect

Probabilities of causation are proven to be critical in modern decision-making. This paper deals with the problem of estimating the probabilities of causation when treatment and effect are not binary. Pearl defined the binary probabilities of causation, such as the probability of necessity and sufficiency (PNS), the probability of sufficiency (PS), and the probability of necessity (PN). Tian and Pearl then derived sharp bounds for these probabilities of causation using experimental and observational data. In this paper, we define and provide theoretical bounds for all types of probabilities of causation with multivalued treatments and effects. We further discuss examples where our bounds guide practical decisions and use simulation studies to evaluate how informative the bounds are for various data combinations.