  Counterfactual explanations (CE) are the de facto method of providing insight
and interpretability in black-box decision-making models by identifying
alternative input instances that lead to different outcomes. This paper extends
the concept of CE to a distributional context, broadening the scope from
individual data points to entire input and output distributions, named
distributional counterfactual explanation (DCE). In DCE, we take the
stakeholder's perspective and shift focus to analyzing the distributional
properties of the factual and counterfactual, drawing parallels to the
classical approach of assessing individual instances and their resulting
decisions. We leverage optimal transport (OT) to frame a chance-constrained
optimization problem, aiming to derive a counterfactual distribution that
closely aligns with its factual counterpart, substantiated by statistical
confidence. Our proposed optimization method, Discount, strategically balances
this confidence in both the input and output distributions. This algorithm is
accompanied by an analysis of its convergence rate. The efficacy of our
proposed method is substantiated through a series of quantitative and
qualitative experiments, highlighting its potential to provide deep insights
into decision-making models.
