# Post-hoc Stochastic Concept Bottleneck Models

arXiv: https://arxiv.org/abs/2510.08219 · HTML: https://mlsanity.com/paper/2510.08219/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 27 references that are in the map.
- First posted on arXiv 9 Oct 2025.

## Builds on (most cited first, 16 of 16)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Towards A Rigorous Science of Interpretable Machine Learning](https://mlsanity.com/paper/1702.08608/) (2017)
- [Concept Bottleneck Models](https://mlsanity.com/paper/2007.04612/) (2020)
- [The Mythos of Model Interpretability](https://mlsanity.com/paper/1606.03490/) (2016)
- [Post-hoc Concept Bottleneck Models](https://mlsanity.com/paper/2205.15480/) (2022)
- [Label-Free Concept Bottleneck Models](https://mlsanity.com/paper/2304.06129/) (2023)
- [Promises and Pitfalls of Black-Box Concept Learning Models](https://mlsanity.com/paper/2106.13314/) (2021)
- [Do Concept Bottleneck Models Learn as Intended?](https://mlsanity.com/paper/2105.04289/) (2021)
- [Interactive Concept Bottleneck Models](https://mlsanity.com/paper/2212.07430/) (2022)
- [A Closer Look at the Intervention Procedure of Concept Bottleneck Models](https://mlsanity.com/paper/2302.14260/) (2023)
- [Stochastic Concept Bottleneck Models](https://mlsanity.com/paper/2406.19272/) (2024)
- [Learning to Receive Help: Intervention-Aware Concept Embedding Models](https://mlsanity.com/paper/2309.16928/) (2023)
- [Beyond Concept Bottleneck Models: How to Make Black Boxes Intervenable?](https://mlsanity.com/paper/2401.13544/) (2024)
- [Measuring Leakage in Concept-Based Methods: An Information Theoretic Approach](https://mlsanity.com/paper/2504.09459/) (2025)
- [Improving Intervention Efficacy via Concept Realignment in Concept Bottleneck Models](https://mlsanity.com/paper/2405.01531/) (2024)
- [Interpretable Reward Modeling with Active Concept Bottlenecks](https://mlsanity.com/paper/2507.04695/) (2025)

## Abstract (arXiv)

Concept Bottleneck Models (CBMs) are interpretable models that predict the target variable through high-level human-understandable concepts, allowing users to intervene on mispredicted concepts to adjust the final output. While recent work has shown that modeling dependencies between concepts can improve CBM performance, especially under interventions, such approaches typically require retraining the entire model, which may be infeasible when access to the original data or compute is limited. In this paper, we introduce Post-hoc Stochastic Concept Bottleneck Models (PSCBMs), a lightweight method that augments any pre-trained CBM with a multivariate normal distribution over concepts by adding only a small covariance-prediction module, without retraining the backbone model. We propose two training strategies and show on real-world data that PSCBMs consistently match or improve both concept and target accuracy over standard CBMs at test time. Furthermore, we show that due to the modeling of concept dependencies, PSCBMs perform much better than CBMs under interventions, while remaining far more efficient than retraining a similar stochastic model from scratch.