# Generating Likely Counterfactuals Using Sum-Product Networks

arXiv: https://arxiv.org/abs/2401.14086 · HTML: https://mlsanity.com/paper/2401.14086/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 73 references that are in the map.
- First posted on arXiv 25 Jan 2024.

## Builds on (most cited first, 30 of 31)

- [Counterfactual Explanations without Opening the Black Box: Automated Decisions and the GDPR](https://mlsanity.com/paper/1711.00399/) (2017)
- [Explaining Machine Learning Classifiers through Diverse Counterfactual Explanations](https://mlsanity.com/paper/1905.07697/) (2019)
- [Actionable Recourse in Linear Classification](https://mlsanity.com/paper/1809.06514/) (2018)
- [Algorithmic Recourse: from Counterfactual Explanations to Interventions](https://mlsanity.com/paper/2002.06278/) (2020)
- [FACE: Feasible and Actionable Counterfactual Explanations](https://mlsanity.com/paper/1909.09369/) (2019)
- [A Survey on the Explainability of Supervised Machine Learning](https://mlsanity.com/paper/2011.07876/) (2020)
- [Trustworthy AI: From Principles to Practices](https://mlsanity.com/paper/2110.01167/) (2021)
- [Learning Model-Agnostic Counterfactual Explanations for Tabular Data](https://mlsanity.com/paper/1910.09398/) (2019)
- [Preserving Causal Constraints in Counterfactual Explanations for Machine Learning Classifiers](https://mlsanity.com/paper/1912.03277/) (2019)
- [Benchmarking and Survey of Explanation Methods for Black Box Models](https://mlsanity.com/paper/2102.13076/) (2021)
- [A survey of algorithmic recourse: definitions, formulations, solutions, and prospects](https://mlsanity.com/paper/2010.04050/) (2020)
- [Efficient Search for Diverse Coherent Explanations](https://mlsanity.com/paper/1901.04909/) (2019)
- [CARLA: A Python Library to Benchmark Algorithmic Recourse and Counterfactual Explanation Algorithms](https://mlsanity.com/paper/2108.00783/) (2021)
- [Evaluating Robustness of Counterfactual Explanations](https://mlsanity.com/paper/2103.02354/) (2021)
- [When Deep Learning Meets Polyhedral Theory: A Survey](https://mlsanity.com/paper/2305.00241/) (2023)
- [Optimal Counterfactual Explanations in Tree Ensembles](https://mlsanity.com/paper/2106.06631/) (2021)
- [OMLT: Optimization & Machine Learning Toolkit](https://mlsanity.com/paper/2202.02414/) (2022)
- [Convex Density Constraints for Computing Plausible Counterfactual Explanations](https://mlsanity.com/paper/2002.04862/) (2020)
- [Diffeomorphic Counterfactuals with Generative Models](https://mlsanity.com/paper/2206.05075/) (2022)
- [SPFlow: An Easy and Extensible Library for Deep Probabilistic Learning using Sum-Product Networks](https://mlsanity.com/paper/1901.03704/) (2019)
- [Bayesian Learning of Sum-Product Networks](https://mlsanity.com/paper/1905.10884/) (2019)
- [Formalising the Robustness of Counterfactual Explanations for Neural Networks](https://mlsanity.com/paper/2208.14878/) (2022)
- [Scaling Guarantees for Nearest Counterfactual Explanations](https://mlsanity.com/paper/2010.04965/) (2020)
- [Provably Robust and Plausible Counterfactual Explanations for Neural Networks via Robust Optimisation](https://mlsanity.com/paper/2309.12545/) (2023)
- [Finding Regions of Counterfactual Explanations via Robust Optimization](https://mlsanity.com/paper/2301.11113/) (2023)
- [Deep Structured Mixtures of Gaussian Processes](https://mlsanity.com/paper/1910.04536/) (2019)
- [Keep Your Friends Close and Your Counterfactuals Closer: Improved Learning From Closest Rather Than Plausible Counterfactual Explanations in an Abstract Setting](https://mlsanity.com/paper/2205.05515/) (2022)
- [Achieving Diversity in Counterfactual Explanations: a Review and Discussion](https://mlsanity.com/paper/2305.05840/) (2023)
- [CeFlow: A Robust and Efficient Counterfactual Explanation Framework for Tabular Data using Normalizing Flows](https://mlsanity.com/paper/2303.14668/) (2023)
- [Nonconvex piecewise linear functions: Advanced formulations and simple modeling tools](https://mlsanity.com/paper/1708.00050/) (2017)

## Built on by (most cited first, 6 of 6)

- [PAR: Plausibility-aware Amortized Recourse Generation](https://mlsanity.com/paper/2601.17309/) (2026)
- [Counterfactual Explanations in Medical Imaging: Exploring SPN-Guided Latent Space Manipulation](https://mlsanity.com/paper/2507.19368/) (2025)
- [Constraint-Reduced MILP with Local Outlier Factor Modeling for Plausible Counterfactual Explanations in Credit Approval](https://mlsanity.com/paper/2509.19504/) (2025)
- [Tractable Probabilistic Models for Investment Planning](https://mlsanity.com/paper/2511.13888/) (2025)
- [Diverse and Plausible Algorithmic Recourse via Tractable Recourse Distributions](https://mlsanity.com/paper/2608.04677/) (2026)
- [A Compositional Theory of Curvature in Probabilistic Circuits](https://mlsanity.com/paper/2608.12869/) (2026)

## Abstract (arXiv)

The need to explain decisions made by AI systems is driven by both recent regulation and user demand. The decisions are often explainable only post hoc. 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). We use a Sum-Product Network (SPN) to estimate the likelihood of a counterfactual. To achieve that, we propose an MIO formulation of an SPN, which can be of independent interest. The source code with examples is available at https://github.com/Epanemu/LiCE.