# Causal Neural Probabilistic Circuits

arXiv: https://arxiv.org/abs/2603.01372 · HTML: https://mlsanity.com/paper/2603.01372/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 62 references that are in the map.
- First posted on arXiv 2 Mar 2026.

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

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Benchmarking Neural Network Robustness to Common Corruptions and Perturbations](https://mlsanity.com/paper/1903.12261/) (2019)
- [A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks](https://mlsanity.com/paper/1610.02136/) (2016)
- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [Enhancing The Reliability of Out-of-distribution Image Detection in Neural Networks](https://mlsanity.com/paper/1706.02690/) (2017)
- [Concept Bottleneck Models](https://mlsanity.com/paper/2007.04612/) (2020)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [Post-hoc Concept Bottleneck Models](https://mlsanity.com/paper/2205.15480/) (2022)
- [Label-Free Concept Bottleneck Models](https://mlsanity.com/paper/2304.06129/) (2023)
- [Learning Bayesian Networks: The Combination of Knowledge and Statistical Data](https://mlsanity.com/paper/1302.6815/) (2013)
- [Right for the Right Concept: Revising Neuro-Symbolic Concepts by Interacting with their Explanations](https://mlsanity.com/paper/2011.12854/) (2020)
- [A Differential Approach to Inference in Bayesian Networks](https://mlsanity.com/paper/1301.3847/) (2013)
- [Interactive Concept Bottleneck Models](https://mlsanity.com/paper/2212.07430/) (2022)
- [VLG-CBM: Training Concept Bottleneck Models with Vision-Language Guidance](https://mlsanity.com/paper/2408.01432/) (2024)
- [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 Intervene on Concept Bottlenecks](https://mlsanity.com/paper/2308.13453/) (2023)
- [Causal Concept Graph Models: Beyond Causal Opacity in Deep Learning](https://mlsanity.com/paper/2405.16507/) (2024)
- [A Unified Approach for Learning the Parameters of Sum-Product Networks](https://mlsanity.com/paper/1601.00318/) (2016)
- [Causally Reliable Concept Bottleneck Models](https://mlsanity.com/paper/2503.04363/) (2025)
- [Causal Inference Using Tractable Circuits](https://mlsanity.com/paper/2202.02891/) (2022)
- [An Advance on Variable Elimination with Applications to Tensor-Based Computation](https://mlsanity.com/paper/2002.09320/) (2020)
- [Tractable Boolean and Arithmetic Circuits](https://mlsanity.com/paper/2202.02942/) (2022)
- [Neural Probabilistic Circuits: Enabling Compositional and Interpretable Predictions through Logical Reasoning](https://mlsanity.com/paper/2501.07021/) (2025)
- [Understanding and Improving Adversarial Robustness of Neural Probabilistic Circuits](https://mlsanity.com/paper/2509.20549/) (2025)

## Abstract (arXiv)

Concept Bottleneck Models (CBMs) enhance the interpretability of end-to-end neural networks by introducing a layer of concepts and predicting the class label from the concept predictions. A key property of CBMs is that they support interventions, i.e., domain experts can correct mispredicted concept values at test time to improve the final accuracy. However, typical CBMs apply interventions by overwriting only the corrected concept while leaving other concept predictions unchanged, which ignores causal dependencies among concepts. To address this, we propose the Causal Neural Probabilistic Circuit (CNPC), which combines a neural attribute predictor with a causal probabilistic circuit compiled from a causal graph. This circuit supports exact, tractable causal inference that inherently respects causal dependencies. Under interventions, CNPC models the class distribution based on a Product of Experts (PoE) that fuses the attribute predictor's predictive distribution with the interventional marginals computed by the circuit. We theoretically characterize the compositional interventional error of CNPC w.r.t. its modules and identify conditions under which CNPC closely matches the ground-truth interventional class distribution. Experiments on five benchmark datasets in both in-distribution and out-of-distribution settings show that, compared with five baseline models, CNPC achieves higher task accuracy across different numbers of intervened attributes.