Explainability of decisions made by AI systems is driven by both recent regulation and user demand. These decisions are often explainable only \emph{post hoc}, after the fact. In counterfactual explanations, one may ask what constitutes the best counterfactual explanation. Clearly, multiple criteria must be taken into account, although "distance from the sample" is a key criterion. Recent methods that consider the plausibility of a counterfactual seem to sacrifice this original objective. Here, we present a system that provides high-likelihood explanations that are, at the same time, close and sparse. We show that the search for the most likely explanations satisfying many common desiderata for counterfactual explanations can be modeled using mixed-integer optimization (MIO). In the process, we propose an MIO formulation of a Sum-Product Network (SPN) and use the SPN to estimate the likelihood of a counterfactual, which can be of independent interest.

翻译：人工智能系统决策的可解释性既受到近期法规的推动，也源于用户需求。这些决策通常只能在事后进行解释。在反事实解释中，人们可能会问什么构成了最佳的反事实解释。显然，尽管"与样本的距离"是关键标准，但必须考虑多重准则。近期考虑反事实合理性的方法似乎牺牲了这一原始目标。本文提出一种系统，能够同时提供既接近原样本又具有稀疏性的高似然解释。我们证明，寻找满足反事实解释多种常见需求的最可能解释问题，可以通过混合整数优化（MIO）建模。在此过程中，我们提出了和积网络（SPN）的MIO表述，并利用SPN估计反事实的似然度——该方法本身也具有独立研究价值。