# Ranking Counterfactual Explanations

arXiv: https://arxiv.org/abs/2503.15817 · HTML: https://mlsanity.com/paper/2503.15817/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 25 references that are in the map.
- First posted on arXiv 20 Mar 2025.

## Builds on (most cited first, 15 of 15)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Explanation in Artificial Intelligence: Insights from the Social Sciences](https://mlsanity.com/paper/1706.07269/) (2017)
- [Counterfactual Explanations without Opening the Black Box: Automated Decisions and the GDPR](https://mlsanity.com/paper/1711.00399/) (2017)
- [European Union regulations on algorithmic decision-making and a "right to explanation"](https://mlsanity.com/paper/1606.08813/) (2016)
- [Explanations based on the Missing: Towards Contrastive Explanations with Pertinent Negatives](https://mlsanity.com/paper/1802.07623/) (2018)
- [Adversarial Attacks and Defences: A Survey](https://mlsanity.com/paper/1810.00069/) (2018)
- [Local Rule-Based Explanations of Black Box Decision Systems](https://mlsanity.com/paper/1805.10820/) (2018)
- [Abduction-Based Explanations for Machine Learning Models](https://mlsanity.com/paper/1811.10656/) (2018)
- [Good Counterfactuals and Where to Find Them: A Case-Based Technique for Generating Counterfactuals for Explainable AI (XAI)](https://mlsanity.com/paper/2005.13997/) (2020)
- [Explanations for Monotonic Classifiers](https://mlsanity.com/paper/2106.00154/) (2021)
- [On Relating 'Why?' and 'Why Not?' Explanations](https://mlsanity.com/paper/2012.11067/) (2020)
- [A Symbolic Approach for Counterfactual Explanations](https://mlsanity.com/paper/2206.09638/) (2022)
- [Causal Generative Explainers using Counterfactual Inference: A Case Study on the Morpho-MNIST Dataset](https://mlsanity.com/paper/2401.11394/) (2024)
- [Axiomatic Characterisations of Sample-based Explainers](https://mlsanity.com/paper/2408.04903/) (2024)

## Abstract (arXiv)

AI-driven outcomes can be challenging for end-users to understand. Explanations can address two key questions: "Why this outcome?" (factual) and "Why not another?" (counterfactual). While substantial efforts have been made to formalize factual explanations, a precise and comprehensive study of counterfactual explanations is still lacking. This paper proposes a formal definition of counterfactual explanations, proving some properties they satisfy, and examining the relationship with factual explanations. Given that multiple counterfactual explanations generally exist for a specific case, we also introduce a rigorous method to rank these counterfactual explanations, going beyond a simple minimality condition, and to identify the optimal ones. Our experiments with 12 real-world datasets highlight that, in most cases, a single optimal counterfactual explanation emerges. We also demonstrate, via three metrics, that the selected optimal explanation exhibits higher representativeness and can explain a broader range of elements than a random minimal counterfactual. This result highlights the effectiveness of our approach in identifying more robust and comprehensive counterfactual explanations.