Due to user demand and recent regulation (GDPR, AI Act), decisions made by AI systems need to be explained. These decisions are often explainable only post hoc, where counterfactual explanations are popular. The question of what constitutes the best counterfactual explanation must consider multiple aspects, where "distance from the sample" is the most common. We argue that this requirement frequently leads to explanations that are unlikely and, therefore, of limited value. Here, we present a system that provides high-likelihood explanations. 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. A numerical comparison against several methods for generating counterfactual explanations is provided.

翻译：由于用户需求及近期法规（GDPR、AI法案），人工智能系统所作出的决策需要得到解释。这些决策通常只能通过事后解释的方式说明，其中反事实解释较为流行。关于何为最佳反事实解释的问题必须考虑多个维度，其中"样本距离"是最常见的考量标准。我们认为这一要求常导致解释内容缺乏可信度，因此价值有限。本文提出一种能提供高似然解释的系统。我们论证，满足反事实解释常见需求的最优解释搜索问题，可通过混合整数优化（MIO）建模。在此过程中，我们提出求和-积网络（SPN）的MIO表达式，并利用SPN估计反事实的似然度——这一方法本身亦具有独立研究价值。最后，我们与多种生成反事实解释的方法进行了数值比较。