Causal Neural Probabilistic Circuits

Weixin Chen; Han Zhao

cs.LGMar 2, 2026

Causal Neural Probabilistic Circuits

Authors: Weixin Chen, Han Zhao

Organizations: University of Illinois Urbana-Champaign

Abstract

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.

Figures & tables

Figure 1: Model architectures and intervention procedures of a CBM and a CNPC. A typical CBM (top-left module + top-right module) performs the intervention on $A_{1}$ by replacing the neural network’s predictions for $A_{1}$ with the ground-truth distribution, while leaving the predictions for other attributes unchanged. In contrast, CNPC (top-left module + bottom module) combines a neural network with a causal PC compiled from a causal graph, and approximates the interventional class distribution based on a PoE with a balancing weight $\alpha$ that fuses complementary information from the two modules.

Figure 2: A causal PC that compiles the causal graph $V_{2}\leftarrow V_{1}\rightarrow V_{3}$ .

Figure 3: Task accuracy of all models in the benign setting on the Asia, Sachs, MNISTAdd, and CelebA datasets under varying numbers of attribute interventions. All results are averaged across three random seeds.

Figure 4: Task accuracy of all models in OOD settings on the MNISTAdd and CelebA datasets under varying numbers of attribute interventions. All results are averaged across three random seeds.

Figure 5: Task accuracy of CNPC in both benign and OOD settings with $\alpha$ varying from 0.0 to 1.0 in increments of 0.1. Left: Performance on MNISTAdd with one intervened attribute. Middle/Right: Performance on CelebA with two/four intervened attributes. All results are averaged across three random seeds.

Panel of Figure 6 — Figure 6: Attribute accuracy ( left ) and task accuracy ( right ) of all models on cMNISTAdd.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Step	Variable	Factors	Variables	New
	eliminated	used	involved	factor
1	$V_{2}$	$\phi_{V_{2}}(V_{1},V_{2}),\ \lambda_{V_{2}}(V_{2})$	$V_{1},\ V_{2}$	$\tau_{1}(V_{1})$
2	$V_{3}$	$\phi_{V_{3}}(V_{1},V_{3}),\ \lambda_{V_{3}}(V_{3})$	$V_{1},\ V_{3}$	$\tau_{2}(V_{1})$
3	$V_{1}$	$\phi_{V_{1}}(V_{1}),\ \tau_{1}(V_{1}),\ \tau_{2}(V_{1}),\ \lambda_{V_{1}}(V_{1})$	$V_{1}$	$f$

Appendix

Table 1: A run of VE for the network polynomial with the elimination order

[V_{2},\ V_{3},\ V_{1}]

Panel of Figure 7 — Figure 7: Causal graphs over attributes $A_{1:K}$ and the class label $Y$ . Left: The causal graph for MNISTAdd, with attributes $A_{1}$ : the first digit and $A_{2}$ : the second digit. Right: The causal graph for CelebA, with attributes $A_{1}$ : Attractive, $A_{2}$ : Mouth_Slightly_Open, $A_{3}$ : Smiling, $A_{4}$ : Wearing_Lipstick, and $A_{5}$ : High_Cheekbones.

Figure 8: Instances from the cMNISTAdd dataset. Top: Examples from the training set. Bottom: Examples from the test set.

Figure 9: Attribute accuracy of CNPC and baseline models in the benign setting on the Asia, Sachs, MNISTAdd, and CelebA datasets under varying numbers of attribute interventions. All results are averaged across three random seeds.

Figure 10: Attribute accuracy of CNPC and baseline models in OOD settings on the MNISTAdd and CelebA datasets under varying numbers of attribute interventions. All results are averaged across three random seeds.

Concept Bottleneck Models

Concept Bottleneck Models route predictions through human-understandable concepts that people can inspect, correct, or intervene on.

See all 87 papers

Jun 5, 2025

Towards Reasonable Concept Bottleneck Models

We propose a novel, flexible, and efficient framework for designing Concept Bottleneck Models (CBMs) that enables practitioners to explicitly encode and extend their prior knowledge and beliefs about the concept-concept (

C-C

) and concept-task (

C \to Y

) relationships within the model's reasoning when making predictions. The resulting

\textbf{C}

oncept

\textbf{REA}

soning

\textbf{M}

odels (CREAMs) architecturally encode arbitrary types of

C-C

relationships such as mutual exclusivity, hierarchical associations, and/or correlations, as well as potentially sparse

C \to Y

relationships. Moreover, CREAM can optionally incorporate a regularized side-channel to complement the potentially {incomplete concept sets}, achieving competitive task performance while encouraging predictions to be concept-grounded. To evaluate CBMs in such settings, we introduce a

C \to Y

agnostic metric that quantifies interpretability when predictions partially rely on the side-channel. In our experiments, we show that, without additional computational overhead, CREAM models support efficient interventions, can avoid concept leakage, and achieve black-box-level performance under missing concepts. We further analyze how an optional side-channel affects interpretability and intervenability. Importantly, the side-channel enables CBMs to remain effective even in scenarios where only a limited number of concepts are available.

Nektarios Kalampalikis, Kavya Gupta, Georgi Vitanov +1

Department of Computer Science Saarland University, Germany

Concept Bottleneck Models Interpretable ML

Ask PDF

May 20, 2026

SynCB: A Synergy Concept-Based Model with Dynamic Routing Between Concepts and Complementary Neural Branches

Concept-based (CB) models provide interpretability and support test-time human intervention, while standard neural networks (NN) offer strong task performance but little transparency. Prior work has explored hybrid formulations that integrate concepts and additional representations to improve accuracy, often at the cost of human interventions. We introduce the Synergy Concept-Based Model (SynCB) framework, that combines a CB branch with a complementary neural branch, and a trainable routing module that dynamically selects which branch to use for each input. Unlike prior models, which fuse residual and concept-based predictions, SynCB keeps the two branches distinct and coordinates them through the routing module. Moreover, both branches are learned jointly, allowing information sharing between the complementary neural branch and CB branches through their common backbone. To improve responsiveness to interventions, we further introduce a test-time intervention policy and a corresponding loss. Across five datasets and CB benchmarks, SynCB consistently achieves higher task accuracy while remaining more responsive to human interventions, surpassing the full neural baseline by up to 3.9 percentage points and exceeding the strongest competitor in intervention performance by up to 6.43 percentage points.

Tores Julie, Sun Rémy, Sassatelli Lucile +3

1Université Côte d’Azur, CNRS, Inria, I3S · 2Université Côte d’Azur, Inria, CNRS, I3S · 3Université Côte d’Azur, CNRS, I3S +1

Concept Bottleneck Models

Ask PDF

Oct 5, 2026

When Are Concept Bottleneck Model Explanations Faithful and Compact?

Concept bottleneck models (CBMs) are neural classifiers that allow to explain their decisions via high-level concepts, potentially enabling understanding, steering and debugging. However, their explanations are often derived heuristically. Building on formal explainability, we argue they should also be faithful, i.e., not misreport which concepts actually matter. We show that, for widespread CBM architectures, including recent VLM-based variants, faithful explanations must include all concepts in the bottleneck, compromising interpretability when this is large. This result applies to both heuristic and faithful-by-construction formal explanations. To encourage the existence of compact faithful explanations, we suggest i) modeling concepts probabilistically as binary or categorical random variables (rather than logits), and ii) employing per-concept training-time sparsification via group lasso (rather than regular elastic net). We also extend algorithms from formal explainability to CBMs, and show they outperform natural heuristics in terms of guarantees and explanation size. Overall, our work warns against naive interpretability claims and provides formal conditions and practical strategies for ensuring CBMs are as interpretable as advertised.

Stefano Teso, Emanuele Marconato, Steve Azzolin +1

CIMeC & DISI University of Trento · Department of Mathematical Sciences University of Copenhagen · DISI University of Trento +1

Explainable Artificial Intelligence Concept Bottleneck Models

Ask PDF

Abstract

Concept Bottleneck Models

Explore similar work

Towards Reasonable Concept Bottleneck Models

SynCB: A Synergy Concept-Based Model with Dynamic Routing Between Concepts and Complementary Neural Branches

When Are Concept Bottleneck Model Explanations Faithful and Compact?