Wesummarize all unidentifiable situations that are discovered in the causal inference literature. Then, we develop a constrained optimization problem forbounding thePCfairness, whichismotivatedbythemethod proposed in[2]forbounding confounded causaleffects. Thekeyideaistoparameterize thecausal model using so-called response-function variables, whose distribution captures all randomness encoded in the causal model, so that we can explicitly traverse all possible causal models to find thetightest possible bounds.

In this paper, we develop a framework for measuring different causality-based fairness.

A recent trend of fair machine learning is to define fairness as causality-based notions which concern the causal connection between protected attributes and decisions. However, one common challenge of all causality-based fairness notions is identifiability, i.e., whether they can be uniquely measured from observational data, which is a critical barrier to applying these notions to real-world situations. In this paper, we develop a framework for measuring different causality-based fairness. We propose a unified definition that covers most of previous causality-based fairness notions, namely the path-specific counterfactual fairness (PC fairness). Based on that, we propose a general method in the form of a constrained optimization problem for bounding the path-specific counterfactual fairness under all unidentifiable situations. Experiments on synthetic and real-world datasets show the correctness and effectiveness of our method.