# Probabilistically Plausible Counterfactual Explanations with Normalizing Flows

arXiv: https://arxiv.org/abs/2405.17640 · HTML: https://mlsanity.com/paper/2405.17640/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 33 references that are in the map.
- First posted on arXiv 27 May 2024.

## Builds on (most cited first, 26 of 26)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Counterfactual Explanations without Opening the Black Box: Automated Decisions and the GDPR](https://mlsanity.com/paper/1711.00399/) (2017)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Explaining Machine Learning Classifiers through Diverse Counterfactual Explanations](https://mlsanity.com/paper/1905.07697/) (2019)
- [Explanations based on the Missing: Towards Contrastive Explanations with Pertinent Negatives](https://mlsanity.com/paper/1802.07623/) (2018)
- [FACE: Feasible and Actionable Counterfactual Explanations](https://mlsanity.com/paper/1909.09369/) (2019)
- [Interpretable Counterfactual Explanations Guided by Prototypes](https://mlsanity.com/paper/1907.02584/) (2019)
- [Neural Oblivious Decision Ensembles for Deep Learning on Tabular Data](https://mlsanity.com/paper/1909.06312/) (2019)
- [Counterfactual Explanations and Algorithmic Recourses for Machine Learning: A Review](https://mlsanity.com/paper/2010.10596/) (2020)
- [Preserving Causal Constraints in Counterfactual Explanations for Machine Learning Classifiers](https://mlsanity.com/paper/1912.03277/) (2019)
- [Efficient Search for Diverse Coherent Explanations](https://mlsanity.com/paper/1901.04909/) (2019)
- [Diffusion Visual Counterfactual Explanations](https://mlsanity.com/paper/2210.11841/) (2022)
- [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)
- [Diffusion Models for Counterfactual Explanations](https://mlsanity.com/paper/2203.15636/) (2022)
- [GANterfactual - Counterfactual Explanations for Medical Non-Experts using Generative Adversarial Learning](https://mlsanity.com/paper/2012.11905/) (2020)
- [On the computation of counterfactual explanations -- A survey](https://mlsanity.com/paper/1911.07749/) (2019)
- [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)
- [Provably Robust and Plausible Counterfactual Explanations for Neural Networks via Robust Optimisation](https://mlsanity.com/paper/2309.12545/) (2023)
- [A Few Good Counterfactuals: Generating Interpretable, Plausible and Diverse Counterfactual Explanations](https://mlsanity.com/paper/2101.09056/) (2021)
- [Plausible Counterfactuals: Auditing Deep Learning Classifiers with Realistic Adversarial Examples](https://mlsanity.com/paper/2003.11323/) (2020)
- [CeFlow: A Robust and Efficient Counterfactual Explanation Framework for Tabular Data using Normalizing Flows](https://mlsanity.com/paper/2303.14668/) (2023)
- [Hierarchical Variational Autoencoder for Visual Counterfactuals](https://mlsanity.com/paper/2102.00854/) (2021)

## Built on by (most cited first, 11 of 11)

- [Perfect Counterfactuals in Imperfect Worlds: Modelling Noisy Implementation of Actions in Sequential Algorithmic Recourse](https://mlsanity.com/paper/2410.02273/) (2024)
- [Uncertainty Quantification as a Principled Foundation for Explainable Artificial Intelligence: A Case Study of Counterfactual Explanations](https://mlsanity.com/paper/2502.17007/) (2025)
- [DiCoFlex: Model-agnostic diverse counterfactuals with flexible control](https://mlsanity.com/paper/2505.23700/) (2025)
- [Synthesising Counterfactual Explanations via Label-Conditional Gaussian Mixture Variational Autoencoders](https://mlsanity.com/paper/2510.04855/) (2025)
- [HyConEx: Hypernetwork classifier with counterfactual explanations for tabular data](https://mlsanity.com/paper/2503.12525/) (2025)
- [Actionable and diverse counterfactual explanations incorporating domain knowledge and plausibility constraints](https://mlsanity.com/paper/2511.20236/) (2025)
- [CounterFlowNet: From Minimal Changes to Meaningful Counterfactual Explanations](https://mlsanity.com/paper/2602.17244/) (2026)
- [A Probabilistic Consensus-Driven Approach for Robust Counterfactual Explanations](https://mlsanity.com/paper/2604.17494/) (2026)
- [Density-Guided Robust Counterfactual Explanations on Tabular Data under Model Multiplicity](https://mlsanity.com/paper/2605.30901/) (2026)
- [CEL: Comprehensive Counterfactual Explanations Library and Benchmark](https://mlsanity.com/paper/2607.22045/) (2026)
- [Challenges in Evaluating Explanation Methods for Static and Evolving Data](https://mlsanity.com/paper/2608.06351/) (2026)

## Abstract (arXiv)

We present PPCEF, a novel method for generating probabilistically plausible counterfactual explanations (CFs). PPCEF advances beyond existing methods by combining a probabilistic formulation that leverages the data distribution with the optimization of plausibility within a unified framework. Compared to reference approaches, our method enforces plausibility by directly optimizing the explicit density function without assuming a particular family of parametrized distributions. This ensures CFs are not only valid (i.e., achieve class change) but also align with the underlying data's probability density. For that purpose, our approach leverages normalizing flows as powerful density estimators to capture the complex high-dimensional data distribution. Furthermore, we introduce a novel loss that balances the trade-off between achieving class change and maintaining closeness to the original instance while also incorporating a probabilistic plausibility term. PPCEF's unconstrained formulation allows for efficient gradient-based optimization with batch processing, leading to orders of magnitude faster computation compared to prior methods. Moreover, the unconstrained formulation of PPCEF allows for the seamless integration of future constraints tailored to specific counterfactual properties. Finally, extensive evaluations demonstrate PPCEF's superiority in generating high-quality, probabilistically plausible counterfactual explanations in high-dimensional tabular settings. This makes PPCEF a powerful tool for not only interpreting complex machine learning models but also for improving fairness, accountability, and trust in AI systems.