Partial counterfactual identification and uplift modeling: theoretical results and real-world assessment

An example of counterfactual statement is "I got no effect since I made no action but something would have happened had I acted". Counterfactuals are used in many fields, ranging from algorithmic recourse [Karimi et al., 2021] to online advertisement and customer relationship management [Li and Pearl, 2019]. Counterfactuals have been formally defined in terms of structural causal models by Pearl [2009]. Nevertheless, since a counterfactual statement cannot be directly observed, the research focuses on estimating or bounding their probability (e.g. the probability that we have an effect given a treatment and no effect else). The probability of some specific counterfactual expressions have been studied in the literature [Tian and Pearl, 2000] because of their relevance in causal decision-making. The probability of necessity (PN) is the probability that an event y would not have occurred in the absence of an action or treatment t, given that y and t in fact occurred. Conversely, the probability of sufficiency (PS) is the probability that event y would have occurred in the presence of an action t, given that both y and t in fact did not occur. Lastly, the probability of necessity and sufficiency (PNS) is the probability that the event y occurs if and only if the event t occurs. In the case of incomplete knowledge about the causal model, identification procedures indicate when and how the probability of counterfactuals can be computed from a combination of observational data, experimental data (i.e.