  The capacity to address counterfactual "what if" inquiries is crucial for
understanding and making use of causal influences. Traditional counterfactual
inference, under Pearls' counterfactual framework, typically depends on having
access to or estimating a structural causal model. Yet, in practice, this
causal model is often unknown and might be challenging to identify. Hence, this
paper aims to perform reliable counterfactual inference based solely on
observational data and the (learned) qualitative causal structure, without
necessitating a predefined causal model or even direct estimations of
conditional distributions. To this end, we establish a novel connection between
counterfactual inference and quantile regression and show that counterfactual
inference can be reframed as an extended quantile regression problem. Building
on this insight, we propose a practical framework for efficient and effective
counterfactual inference implemented with neural networks under a bi-level
optimization scheme. The proposed approach enhances the capacity to generalize
estimated counterfactual outcomes to unseen data, thereby providing an upper
bound on the generalization error. Furthermore, empirical evidence demonstrates
its superior statistical efficiency in comparison to existing methods.
Empirical results conducted on multiple datasets offer compelling support for
our theoretical assertions.
