Causally Reliable Concept Bottleneck Models
Authors: Giovanni De Felice
Arianna Casanova
Francesco De Santis
Silvia Santini
Johannes Schneider
Pietro Barbiero
Alberto Termine
Venue: NeurIPS 2025 poster ()
Keywords: Explainable AI
Concept-based models
Causal reliability
Deep learning
Primary area: social_and_economic_aspects_of_machine_learning
TL;DR: We propose Causally reliable Concept Bottleneck Models (C2BMs), a class of concept-based architectures that enforce reasoning through a bottleneck of concepts structured according to a model of the real-world causal mechanisms.

Abstract: Concept-based models are an emerging paradigm in deep learning that constrains the inference process to operate through human-interpretable variables, facilitating explainability and human interaction. However, these architectures, on par with popular opaque neural models, fail to account for the true causal mechanisms underlying the target phenomena represented in the data. This hampers their ability to support causal reasoning tasks, limits out-of-distribution generalization, and hinders the implementation of fairness constraints. To overcome these issues, we propose Causally reliable Concept Bottleneck Models (C$^2$BMs), a class of concept-based architectures that enforce reasoning through a bottleneck of concepts structured according to a model of the real-world causal mechanisms. We also introduce a pipeline to automatically learn this structure from observational data and unstructured background knowledge (e.g., scientific literature). Experimental evidence suggests that C$^2$BMs are more interpretable, causally reliable, and improve responsiveness to interventions w.r.t. standard opaque and concept-based models, while maintaining their accuracy.

Causally Reliable Concept Bottleneck Models
Giovanni De Felice" Università della Svizzera Italiana giovanni f ch
Arianna Casanova Flores" University of Liechtenstein
Francesco De Santis" Politecnico di Torino
Silvia Santini Università della Svizzera Italiana
Johannes Schneider University of Liechtenstein
Pietro Barbiero' IBM Research
Alberto Termine"
Scuola Universitaria Professionale della Svizzera Italiana, IDSIA
Abstract
Concept-based models are emerging paradigm in deep learning that constrains the inference process to operate through human-interpretablee variables, facilitating explainability and human interaction. However, these architectures, on par with popular opaque neural models, fail to account for the true causal mechanisms underlying the target phenomena represented in the data. This hampers their ability to support causal reasoning tasks, limits out-of aut-of-distribution generalization, and hinders the implementation of fairness constraints. To overcome these issues, we propose Causally reliable Concept Bottleneck Models (C²) class of concept-bass architectures that enforce reasoning through bottleneck of con- cepts structured according to model of the real-world causal mechanisms. We also introduce a pipeline to automatically learn this structure from observational data and unstructured background knowledge (e.g., scientific literature), Experi- mental evidence suggests that BMs are more interpretable, causally reliable, and improve responsiveness interventions w.r.t. standard opaque and concept-basee based models, while maintaining their accuracy.
Introduction
In recent years, interpretable neural models have become more popular, achieving performance similar to powerful opaque Deep Neural Networks (DNNs) (Alvarez Melis & Jaakkola, 2018 Chen et al.
2019 2020). Among these, Concept Bottleneck Models (CBMs) (Koh al.) 2020; Zarlenga et al.
2022 Yuksekgonul et al. 2022 Barbiero et al. 2023) guarantee high expressivity and interpretabilityy by enforcing DNNs to reason through layer of high-level. human- iamaa-interpretabble variables called concepts (e.g., the "color" and "shape" of an object) (Kim et al., 2018 Achtibat et al.) 2023 Fel al. 2023). In CBMs, neural encoder first maps the raw input concepts, forming a semantically transparent intermediate representation that is used by simple decoder for downstream predictions. Beyond transparency, this design allows human experts to intervene on mispredicted concepts at test time to improve downstream task predictions (Fspinosa Zarlenga et al. 2024).
However, like standard DNN architectures, CBMs remain pure associative models (Pearl 2019) their decision-making process reflects statistical correlations within the data rather than real-wordd causal mechanisms. result, they fail to distinguish between spurious correlations and true causal
"Equal contribution.
I Equal senior authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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relationships. Recognizing this distinction is fundamental to achieving reliable scientific understand- ing, supporting causal reasoning for intervention (Pearl 2009; Peters et al., 2017). enabling robust generalization under distributional shifts, and the implementation of fairness constraints (Schölkopf
al. 2021 Wang et al. 2022).
To address these limitations, we propose Causally reliable Concept Bottleneck Models BCs): a class of concept-based architectures that enforce reasoning through "Causal Bottleneck" (Fig. D of concepts structured according to a model of the real-world causal mechanisms underlying data generation. CsB process information as follows. First, a neural encoder extracts set of latent representations from raw data. Then, information flows from latent representations through a given causal graph where each node represents an interpretable variable (e.g.. "smoker", "bronchitis") At inference time, the value of each variable is predicted from its causal parents through an interpretable structural equation, parametrized adaptively by hypernetwork.
Designing a CMBM requires identi- fying domain-relevant concepts and specifying their causal relationships, a process that depends heavily on expert knowledge, which could be scarce, costly, or entirely unavailable in prac- tice. To mitigate this reliance and fa- vor agile deployment across domains, we propose fully automated pipeline (Fig. [ Causal Graph Construction) for instantiating a C²BM, in which the set of relevant concepts and the causal graph are automatically learned from a mixture of data and unstruc- tured background knowledge.
Background Knowledge
Causal Graph construction
Tuberoulosis
g Neural encoder
Smoker
Dyspnea
U23
Bronchiss
Training Inference
Causal Bottleneck
Figure 1: Causally reliable Concept Bottleneck Models (C*)Ms) enforce reasoning through "Causal Bottleneck" aligned with model of real-world causal mechanisms ob- tained from data and background knowledge.
Experimental evidence shows that C2BMs: (i) improve on consistency with real-world causal mechanisms, without compromising accuracy w.r.t. standard DNN models, CBMs, and their extensions (Sec. 5.Th: (ii) improve interventional accuracy on downstream concepts with fewer interventions (.22, (iii) mitigate reliance on spurious correlations (debiasing, Sec. 5.3 (iv)
permit interventions to remove unethical model behavior and meet fairness requirements (Sec.
2
Preliminaries
We introduce the notation and key formalizations underlying standard CBMs and causal modeling. A more detailed background causality is provided in App. A
Concept Bottleneck Models. CBMs Koh et al. 2020) are inte architectures that explain their predictions using high-level interpretable variables called concepts. Standard CBMs decompose prediction into two stages: neural encoder maps the input X to set of intermediate concepts {ViT and decoder predicts the target Y from V. This yields:
P(Y,
P(Y IV)
PVV
decoder
concept encoder
Concept Embedding Models (CEMs) (Zarlenga al. 2022 enhance CBMs by pairing concepts with high-dimensional embeddings of the form P(U X. where U = {U,)C These embeddings are provided to the decoder to predict the target variable Y. enabling the model achieve performance comparable to standard DNN approaches while maintaining semantic interpretability. Critically, traditional decoders rely bipartite structure assumption, wherein all concepts are treated as direct causes of the target, e.g., ==v f(V,.... Vc) for CBMs. This assumption is often overly simplistic for real-world problems. Bringing the reasoning of concept-based architectures closer to real-world mechanisms constitutes the main focus of this work.
Causal Reliability, model M is causally reliable w.r.t. target phenomenon if and only if the structure of M's decision-making making process is consistent with the causal mechanisms underlying T

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(Termine & Primiero 2024). Although state-of-the-art DNNs and concept-based models offer high expressivity, they lack causal reliability.
Structural Causal Models. The standard framework for modeling causal mechanisms is the structural causal model (SCM) [Bareinboim et al. 2022). An SCM .M is tuple (V, U, F, P),
where:
V is a set of C endogenous variables, modeling observable magnitudes of interest;
U is a set of exogenous variables, modeling unobservable magnitudes determined by factors external to V;
F (1i is set of functions such that
Vi = Ji U.) Vi =i= 1,...,
(2)
where PA; C VV V is the set of the endogenous parents of V,, U: C C is an exogenous parent summarizing all the information influencing V2 that is not explicitly represented in V. and the entire set F forms mapping from Ll to
P(U) is a joint probability distribution over U.
Each SCM can be associated with graphical representation in which nodes correspond to the variables, and edges encode the functional relationships specified by F. Here, we focus on SCMs whose associated graph is a directed acyelic graph (DAG) (Pearl) 1995 Zaffalon et al. 2020b). In most cases, the underlying DAG is unknown and must be inferred from observational data, process known as causal discovery (Peters et al., 2017 Zanga et al. 2022). However, methods based solely on observational data cannot generally guarantee the identification of unique DAG (Peters et al.) 2017). The set of candidate DAGs can be refined by incorporating additional information, which we refer to as background knowledge (Andrews et al. [2020] Abdulaal et al., 2023). This can be drawn from a range of sources, such as human experts, structured repositories of information (e.g.. domain ontologies), "unstructured" samples of information (e.g., scientific papers or other documentation).
3 Related works
Traditional concept-based architectures impose strict bipartite structure in which concept neuron activations are assumed to directly cause task outputs (Koh et al. 2020; Yuksekgonul et al. 2022 Kim et al.] 2023] Oikarinen et al. 2023 Yang et al.) 2023 Barbiero et al.) 2023; Vandenhirtz/ et al. 2024). This strong, often unrealistic assumption can lead to misleading explanations. For example, attributing a lung cancer diagnosis to both 'cough' and 'smoker' concept could risk the false interpretation that reducing coughing could reduce cancer risk. Moreover, most CBMs assume independence among concepts, which is unrealistic, (e.g., 'smoke' and 'fire') and prevents improvements in one concept from propagating to related concepts during interventions. Stochastic CBM (SCBM) (Vandenhirtz et al. (024) and Concept Graph Models (Dominici et al.) (2025) attempt to relax this assumption. However, these approaches capture only associations rather than causal relations, making it vulnerable to spurious correlations in the data. To date, no methodology exists for structuring the concept bottleneck according to a reliable causal model.
it ignores natural co-occurrences
Recent approaches like DiConStruct (Moreira et al. 2024). aim to improve this aspect by generating causal graphs linking concepts to opaque DNN predictions. However, DiConStruct is post-hoc method that may misalign with the original DNN's outputs and relies solely on observational data, neglecting background knowledge and resulting in dnter-detemmined causal structures. Other architectures, such as Neural Causal Models (Ke et al. 2019) and Neural Causal Abstractions (Xial & Bareinboim 2024). impose even stronger assumptions, requiring access to either the true causal graph or a low-resolution structural causal model, which are impractical in many cases.
4 Method
In this section, we introduce Causally reliable Concept Bottleneck Models (C2BMs) and the pipeline we propose to fully automate its instantiation, learning, and functioning (Fig.) 2
3

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Badigrounl krowldge
Hitaddo Cancer
DdA
L1M:: RMG
CPOAS kurnig
Concept Discovery
Causal Graph Discovery
Sinching
Lung Cancer
Denolyge
Georyye
Gendtypo
Dippoot
Tor Deposit
LIN
Prang: Doas Genatoo
Brioking
Lung
C² BM
Parametric
Exogenous Variables
Vizag
encoder iK(-)
Baxoking
The Dep-
Unoog
U Ucewocape
Phute
You Day-
Hypernetworks
(+(+)
i-)+)
x2(+)
Youstee
Structural eq.
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parameters
Figure 2: Overview of the BM fully automated pipeline. The pipeline consists of three key blocks: (i) Concept discovery: discovery and labeling of the relevant variables V from background knowledge; (ii) Causal graph discovery: discovery of the causal graph by integrating data and background knowledge; (iii) the CBBM model, comprising a neural encoder and an adaptively parametrized SCM. Once the model is trained, it can support forward and interventional queries about any endogenous variable (e.g., predicting dyspea).
4.1 Causally reliable Concept Bottleneck Models
A BM is concept-based architecture that leverages the formalism of SCMs to structure "causal bottleneck of concepts". More formally, let: (i) X denoting random variable modeling (possibly noisy) input features; (ii) = [V:)e= be set of C semantically meaningful variables modeled as endogenous variables: (iii) G be DAG connecting variables in V. A C²BM is a neural architecture implementing the tuple (g, Me) where:
g(.) is neural encoder modeling probability distribution P(U[X) over set of latent, high-dimensionall embeddings 1/ = {U.). representng the exogenous variables;
Me is parametric SCM (V,4,Foe X)) (see Sec. J. where we assume parametric form for the functions' Specifically, the structure of the functions is determined by the connectivity of G, and the parameters + are predicted from 2/ by a hypernetwork.
The information flowing along C2BM can described as follows (Fig. - right side). First, the values of the exogenous variables 24 are predicted using the exogenous encoder g(-) from X. Then, the information flows along the SCM Me starting from the endogenous sources (predicted from U) down to the sinks. At each subsequent level of the causal graph, the values of each V; are predicted from values its parents PA; based on the relative structural equation f; Fe.
4.2 Model instantiation
To instantiate C²BM, one requires a labeled dataset D annotated for variables in V, as well as DAG capturing the causal relationships among V. However, such resources may be inaccessible, problem-specific, or heavily dependent on human expertise. fully automated pipeline that enables the use of C also in such complex scenarios. Our approach extracts the necessary components from: (i) potentially unlabeled dataset Dz; (ii) potentially unstructured repository of background knowledge K.
address this challenge, we propose

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Our pipeline (see Fig. 2 addresses the following pooblleob::: (i) causal graph construction, which includes concept discovery, concept labeling (Sec. 4.2.J]. and causal graph discovery (Sec. 4.2.2): and (ii) training of the neural parameters of the encoder and the hypernetwork determining the structural equations (Sec. 4.3). In the following, we outline our implementation for each sub-problem. Specifically, building BM' individual prerequisites will be mostly based on prior work. Note that integrating them into a coherent, automated pipeline instead part this paper's contributions.
Remark 4.1. Concepts and causal graph constitute input for CMM which could remain agnostic to how they are obtained, e.g., provided by human experts. Notably, alternative or novel approaches may be employed, provided they solve the same problems (Loula al. 2025)
4.2.1 Concept discovery and labeling
Problem 4.2 (Concept Discovery). Given a dataset of i.i.d. samples Dr {xi and a back- ground knowledge repository K. relative to a task, identify set of relevant variables V.
In the CBM community, automated concept discovery and labeling using Large Language Models (LL.Ms) has become a standard solution when human supervision unavailable (Oikarinen et al.) 2023 Yang et al. 2023 Srivastava et al., 2024, Yamaguchi & Nishida 2025). In our implementation, we follow the fabel-free CBM approach from Oikarinen et al. (2023). where concepts are discovered by querying an LLM for those most relevant to the task. We then apply a filtering procedure to retain only concepts that meet criteria such as brevity, distinctiveness (i.e., not too similar to each other or the target), and presence in the training data.
Once V are selected, we label the dataset with variable annotations D (X,, V: M to supervise concept learning. To do so, we adopt strategy similar to Oikarinen et al. (2023), leveraging pre-trained contrastive vision-language model, such as CLIP (Radford et al., 2021). This projects
both data samples and the discovered concept names into shared embedding space and computes their alignment to generate concept labels. Full implementation details are provided App.
4.2.2 Causal graph discovery
Problem 4.3 (Causal Discovery). Let G* be the true, unknown, graph over set of variables V from which a dataset was generated. The causal discovery problem consists in recovering G from the observed dataset D ZZnga et al. 2022).
As anticipated in Sec. 2 a promising direction for addressing this problem is to combine standard causal discovery algorithms with knowledge- querying. In our pipeline, focus on well- known class of methods that recover an equivalence class of graphs from data, referred to as the Markov equivalence class (MEC) (Spirtes al. 2001] Pearl 2009 Zanga et al. (.22). This class be compactly represented as a Completed Partially Directed Acyelic Graph (CPDAG), an extension of a DAG where edges remain unoriented when there insufficient evidence in the data to infer causal direction. This is desirable property as it prevents incorrect (spurious) orientations based on the data alone. Specifically, we apply the Greedy Equivalence Search (GES) [Chickering 2002) algorithm, which we found performed well empirically (see App. G.3) Then, we leverage à pre- trained LLM to assess each undirected edge in the CPDAG, orienting the ones corresponding to true causal relationships while discarding those originating from spurious correlations. To improve the robustness and generalizability of this approach, we pair the I.IM with Retrieval Augmented Generation (RAG) technique, which is known to reduce hallucinations and can provide problem- specific knowledge. To further improve robustness and performance, we repeat each query 10 times, selecting the most frequent outcome (Wang et al.) 2023). Implementation details are provided in App.(C C
4.3 Structural equations and model training
Problem 4,4 (Learning structural equations). Let & be the set of edges describing causal connections DAG G connecting variables in V. Given D = {(x,, v.)l and E, predict the parameters θ of the structural equations Fe.
5

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We model the structural functions fi E Fe describing the causal mechanisms relating each endoge- nous variable its endogenous parents as weighted linear sums, i.e. for each i:
V
Σ VEEPA
10
where PA, denotes the of endogenous parents of V. For the parameters f.- we do not learn single parameterization; instead, these are adaptively inferred for each different realization X, by a hypernetwork r(-) (Ha al. 2017 Barbiero al. 2023 Debot et al. 2024), based on the values of the exogenous variables U7 and the graph connectivity. In our implementation, we consider separate hypernetworks r-) (e.g., independent DNNs), each taking as input a separate exogenous variable:
y,. r(E,U); (U;) (g()))).
The design in Eq. B and improves both interpretability and expressivity. To aid (mechanistic) interpretability. structural equations take linear form 1Eq.3/ the value of each a weighted linear combination of its parents. The adaptive re-parameterization of the equation's 's weights performed by the hypernetwork r(-) allows the model to also approximate non-linear relationships among en- dogenous variables (see App. D which also includes proof that C2 is a universal approximator, regardless of the underlying causal graph). This idea in line with existing literature on interpretabil- ity (Ribeiro et al. 2016 Alvarez Melis & Jaakkola 2018) and concept-based methods (Barbiero)
ct al. 2023
Model training. The training of CCBM consists of learning the neural parameters of the encoder g(-) and hypernetwork r(-) end- to-end from the input data. We formalize this by modeling the joint conditional distribution P(V,u, θ X,E which factorizes as:
audegenous
equation
exogenous
Figure 3: Probabilistic graphi- (5)
cal model of C2BM inference.
P(V,,, ② X,E) = P(V 1 P
8,
P(U)
X)
where: P(4 X) represents the exogenous encoder g()); P(e) E,44 represents the hypernework r(.) predicting the structural equations' parameters using the given causal connections and the exogenous variables; P(V 24; θ) represents a causally reliable classifier leveraging the structural equations Fe predict the values of the endogenous variables. Under the Markov condition imposed by the CBBM causal graph, the causally-reliable classifier can be re-written as product of independent distributions i.e.,
P(V P()IU:0) = II P(V PA, r(E,U))
where U, g(X), and r(E, = O₁. From the above factorization, derive the C2BM's training objective, which corresponds to maximizing the empirical log-likelihood of the training data:
Φ- arg max
log P(V. PA., Us r(E,4)))
Remark 4.5. We clarify that we do not claim to identify the true structural functions. Instead, C numerically approximates the outcomes as if they were generated by the underlying (unknown) structural equations. This approximation. along with C'BM's DAG, is sufficient to compute reliable interventions, which is a core objective in the concept-based community (Poeta et al., 2023 et al. 2024).
Steinmann
$ 'Root variables are predicted from the exogenous variables 24 using a neural network. Further details are provided in App.
general formalization. In classification setting, concepts assume categorical values; hence Vi represents the probability a concept state activation.
'We refer V. variable allow for
'This idea also aligns with Balke & Pearl (194) and Zaffalon et al. (2020a), where exogenous variables are used represent relationships between endogenous variables when the structural equations unknown.

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Table 1: Task accuracy (%). Task concepts are follows: dysp (Asia), Akt (Sachs), PropCost (Insurance), BP (Alarm), R5Fest (Hailfinder), parity (cMNIST), mouth slightly open (CelebA), Survival (CUBc), pneumothorax (Pneumoth.). refers reduced concept bottlenecks. Methods matching the performance of OpaQNN and showing a significant improvement over the other considered methods are highlighted in bold. Uncertainties represent 2 sample mean a across runs.
MODEE
SUMANTIC
INSURANCE
HaIL FINDER
CMNIST
CELAMA
CUBc
ORagNN
05.83..
12.8..5
12.0
DL24
74.97
00.3
50.0..
02.
F1.B-ia
05.604.0
07 07.
022 02.7axa
72.2-2a
10.90-
71.07..m
00.5442
70.0.s
56.0xa
52 Tass
C'M
71.2...
06.7114
07.3122
70.91s2
00.55 21
71 21.07)..0
96.2119
767
58.12y
528113
CEM
x
65.00..0
00.7116
00.8
TL.321
98.72
T4 74.72..4
06.3ism
80.1..
00.71ax
01.8
SCBM
06.30.:0
18
58.5
05.30sa
064isa
0733u4
14:18..4
73..a
80.5
70.8
00.5..1
5 Experimental evaluations
We evaluate the performance of the proposed CM pipeline. Experiments are conducted across different datasets and settings, allowing for the investigation of the following aspects: classification accuracy (Sec. 5.I.. causal reliability (Sec.5. 5.1). accuracy under ground-truth interventions (Sec. (..
debiasing (Sec. [5.3). and fairness (Sec. (.44) App. provides additional results and ablations.
The considered datasets include both synthetic and real-world benchmarks. As synthetic datasets, we sample 10 points from each of the five following discrete Bayesian networks available from the bnlearn repository (Scutari, 2010): Asia (Lauritzen & Spiegelhalter 1988), Sachs (Sachs et al. 2005), Insurance (Binder et al. [997). Alarm (Beinlich et al. 1989), and Hailfinder (Abramson ct al. 1996). We include eMNIST, variant of the original dataset (LeCun al., (0100) in which the image data are colored according to custom rules. Additionally, we consider three real-world datasets: CelebA (Liu et al. 2015), facial recognition dataset labeled with different binary facial attributes; CUBc, a custom version of the original bird image dataset (He & Peng 2019) from which we select subset of concepts and define new ones to introduce deeper causal relationships; Siim- Pettm--numttoora (You et al. (023),, containing chest X-ray images annotated with a single label indicating the presence of pneumothorax, without additional concepts their annotations. To generate them, we follow the label-free approach outlined in Sec. 1.2.1 Exhaustive details on all
datasets given in App. E
The performance of the proposed pipeline is investigated alongside an opaque neural baseline predicting the task variable only (OPaqNN) and established state-of-the-art (SOTA) concept-based architectures, namely: CBM (Koh et al. 2020), with linear and non-linear decoder; CEM (Zarlenga et al. 2022); and SCBM (Vandenhirtz et al. 2024). Hyperparameters have been selected via an pair based on performance on the validation set. Further
independent search for each dataset-mode details each model's architecture and hyperparameters
be found in App. F Note that all baselines, except for OpaqNN, provide concept- explanations for their predictions and allow concept interventions at test-time. This excludes architectures such as Self-Explainable Neural Networks (Alvarez Melis & Jaakkola (008)) and Concept Whitening (Chen et al., (020) as they do not offer clear mechanism for intervening on their concept bottlenecks. We also excluded other CBM baselines such as Probabilistic CBMs (Kim et al. 22023), ), Post-hoc CBMs (Yuksekgonul et al., 2022). Label-free CBMs (Oikarinen et al. 2023 Yang et al. 2023), all of them share the same limitation of vanilla CBMs and CEMs: the causal graph is fixed and bipartite. Python code for reproducing all experiments is provided alongside the submission as supplementary material.
5.1 Task accuracy and causal reliability
Our initial experiment evaluates task accuracy. For each dataset, we designate a predefined single variable as the prediction task. All models except OpaqNN are trained to predict the task while simultaneously learning to the remaining concepts. Tab. presents the task accuracy for all evaluated models (see G. for concept accuracy). To further assess model expressiveness, we also evaluate task accuracy on modified versions of the Asia and Alarm datasets, where selected concepts (App. E are intentionally removed to create a stronger bottleneck.
7

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Table Structural Hamming distance (App. and number of mistaken edges between true and learned DAG. Reliability of standard flat CBMs is reported for reference. The total number of edges parentheses.
METRIC
AFTER
CMNIST
ASIA
SACHS
INSUR.
ALARM
HAILE
HAMMING
FLAT CBM
1.0
6.5
11.75
36.5
45.0
69.0
CD
0.2
0.7
3.4
6.4
5.4
11.0
CD LLM
0
0.3
1.8
6.3
5.0
11.0
INCORRECT
FLAT CBM
(())
11 (8)
23 (17)
74 (52)
78 (46)
117 (66)
EDGES
CD
(1)
3 (8)
17 (17)
19 (52)
13 (46)
22 (66)
(TRUE EDGES)
CD LLM
(1)
1(8)
7 (17)
18 (52)
10 (46)
22 (66)
BM achieves comparable or higher accuracy to non-causally reliable models (Tab. Our evaluation shows that CM achieves robust accuracy across datasets, matching the performance of the expressive models OpaqueNN and CEM. Notably, as the concept bottleneck is reduced, CM retains expressivity by leveraging exogenous variables propagate residual information from the input. This is in contrast with CBMs implementing a hard bottleneck.
CMBM improves on causal reliability (Tab. 2. C²BM captures a rich causal structure that aligns well with real-world dependencies. We quantitatively assess this alignment by comparing the learned and true causal graphs in synthetic datasets. Tab. Preports two metrics: structural Hamming distance (detailed in App. I. and the number of incorrect edges, computed after causal discovery (CD) and refinement via LLM queries. Metrics for the simplistic graphs from CBMs (all concepts are treated as mutually independent and direct causes the task) are reported for reference. Results indicate that integrating CD with background knowledge produces causal graph that is more accurately aligned with the true structure. Notably, on the Sachs dataset,
the integration of background knowledge enables to correctly identify 10 additional edges w.r.t. CD alone. Detailed ablation studies on causal graph discovery methods and L.MM types are provided in To further validate the quality of the learned causal graph, App. G.2 demonstrates that BM achieves comparable task accuracy using either the learned or the true graph. When considered together, the results in Tab. highlight CBBM's ability to improve causal reliability without compromising expressivity and performance.
App. G.3 G.4 G.5
5.2 Ground-truth interventions
After training all models on the same classification task as in Sec. 5.1 we test their responsiveness to ground-truth interventions, i.e., replacing predicted concepts with ground-truth values 6 This simulates form of human intervention in deployed model. Following each intervention. we compute the average accuracy over all variables (concepts and task) prediction. As for the policy, we intervene on random concepts within progressively deeper levels in the hierarchy defined by the true graph. This constitutes the only intervention policy aligned with real-world causal-effect relationships. When the true graph is unavailable, we use the one generated by pipeline.
CM improves accuracy downstream concepts with fewer interventions (Fig. Our findings, reported in Fig. [4] demonstrate that CBBM achieves higher accuracy improvements with fewer interventions compared to alternative models. This advantage stems from two key properties of CBBM: (i) unlike other baselines that do not account for connections among concepts, interventions on an upstream concept CMB directly influence all downstream nodes, potentially enhancing the predictions of their values; (ii) unlike SCBM, the effects of interventions in BM are restricted to concepts that are causally related, rather than altering concept values due spurious correlations.
*Gronnd -ruth interventions can be seen as special case of causal do fo-interventions (see App. A.1 where variables to their ground-trute values.

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100
100
Asia
Sachs
Insurance
Alarm
88 8 E
94
30
8
80
mam
Hailfinder
CelebA
110
CUBc
Pneumothorax
8
8
25
95
8
8
38
21
80
8
8
75
intervened concepts
Number concepts
intervened concepts
concepts
+ CBM-1 + CBM
+
CEM + SCBM & CFBM
Figure 4: Label accuracy (%) downstream variables (task included) after intervening concepts up progressively deeper levels the graph hierarchy. Summit plots show the difference of C2 accuracy w.r.t. the best-performing baseline. Uncertainties represent 2 sample mean across runs.
5.3 Debiasing
We hypothesize that real-world-alignee causal bottleneck reduce reliance spurious corre- lations. To test this, we use cMNIST, where digit Color is correlated with Parity during training (all the odd digits are green). At test time, the correlation among Color and Parity is reversed (all the even digits are green), introducing a distribution shift that challenges generalization. The neural encoders in all models still struggle with out-of-distribution (OOD) generalization. Therefore, as expected, all models including CMM,, fail to extrapolate correctly, capturing the artificial correlation. However, their enforced reasoning is different. All baselines retain the concept-task connections, perpetuating the color-pane color-parity shortcut. In contrast, CM detects the color -ariity edge through causal discovery but correctly removes via graph refinement with the L.M block. This reflected in large differences in performance after concept interventions, which alleviate (or even remove) the impact of the encoder.
Causal bottlenecks mitigate reliance on spurious cor- relations (Fig. 5) Fig. 5 shows the accuracy on Par- ity after ground- truth interventions on each concept. As expected, color has no effect across all models, confirm- ing the learned bias. Notably, C²BM exhibits the largest improvement when intervening the number concept (achieving ~ 90% accuracy), as its causal structure iso- lates color and strengthens training the correct feature. Although a comprehensive analysis of OOD robustness is beyond the scope this paper, results suggest that the CM pipeline holds promise in improving generalization in biased settings.
CBMr CBM
100
80
60
20
number
Intervened concegt
Figure 5: Biased ColorMNIST dataset. Task accuracy on Parity after ground- truth interventions.
5.4 Fairness
We create a customized CelebA dataset to evaluate the influence of sensitive attributes on the decision- making of the model. Specifically, we consider a hypothetical scenario which an actor with specific physical attribute is required for specific role. However, the hiring manager has strong bias toward Attractive applicants. To model this, we define two custom attributes: Qualified, indicating whether applicant meets the biased hiring criteria, and Should be Hired, which depends on both

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Qualiffed and the task-specific requirement (Pointy Nose). In our fairness analysis, we aim to intervene and remove any such unfair bias.
B'M permit interventions to meet fairness requirements (Tab. 3 We measure the Causal Concept Effect (CaCE) (Goyal et al.] 2019) of Attractive on Should be Hired before and after blocking the only path between them, i.e., performing an intervention on Qualified. Tab. 3 shows that CM is the only model able to successfully remove the influence, achiev- ing a post-intervention CaCE of 0.0%. This difference stems from the model architectures: in CBM, CEM, and SCBM, all con- cepts are directly connected to the one
Table 3: CelebA dataset. Causal Concept Effect (CaCE, % between sensitive concept (Attractive) and a target concept (Should be hired), before and after blocking the path between the two variables with do-interventions.
task, meaning interventions concept cannot block the influence of others. In contrast, C²BM en- forces a structured causal bottleneck allowing for interventions fully override the effects of parent nodes and block information propagation through the intervened node. This highlights C2BM's ability enforce causal fairness by eliminating biased pathways.
CACE METRIC
CBM
CBM
CEM
SCBM
C²BM
REFORE INT.
123
12.5
19.0
30.911.
25.1
AFTER INT.
21.9
11.8
8.2ga.
14.8
0.0g00
6 Conclusions
We presented C²BM, concept-based model advancing prior research by structuring the bottleneck of concepts according to model of causal relations between human-interpretabb variables. By com- bining observational data with background knowledge, CBBMs improves on causal reliability without compromising performance. This offers several additional benefits, e.g., improved interventional accuracy, robustness to spurious correlations, and fairness.
Applications and broader impact. We speculate C has the potential to significantly narrow the hypothesis space in complex scientific domains where even panel of human experts might struggle to identify or exclude plausible hypotheses worth testing. For instance, constructing the hypothesis space to design clinical trials accounting for the influence of environmental conditions on gastrointestinal biochemistry requires deep interdisciplinary knowledge- nott only in environmental science and biochemistry, but also in genetics, microbiology, nutrition science, and epidemiology, among others. In such settings, CBBM may integrate human scientific knowledge across these diverse fields to construct a comprehensive causal graph, thereby supporting experts in systematically excluding hypotheses that are inconsistent with the integrated body of evidence. This can help accelerate interdisciplinary scientific discovery, while reducing experts' cognitive burden.
Limitations. C2BM requires a robust prior knowledge base, access to pre-trained models (LLMs for causal discovery), and well-crafted prompts for querying these LLMs. Biases within the knowledge base or in the observational data can reduce the system's 's reliability (though C2BM still outperforms the selected baselines). Furthermore, the SOTA in causal structural learning currently faces scalability limitations, which also constrain CMM. As these techniques become more scalable and robust, C²BM stands benefit. Finally, encoder embeddings used to construct exogenous variables can move out of distribution, and since the encoder's OOD performance is not guaranteed, the SCM's OOD performance may likewise be affected.
Future works. Future directions include a deeper investigation of OOD generalization with C2BMs, an extensive exploration of their role in causal inference (e.g, counterfactual queries extending O), see App. A). and the identification of optimal intervention policies. Moreover, incorporating PAGs (Zhang 2008) would enable modeling of hidden confounders.
10

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Acknowledgments
This work is supported by the Swiss National Science Foundation (SNSF) through the grant 115121_197244 for the project "PROSELF: Semi-automated Self-Tracking Systems to Improve Personal Productivity" and the Hasler Foundation under the Project ID: 1094-05-15-701 AC and JS acknowledge support from FFF of the University of Liechtenstein grant lbs 24 08. PB acknowl- edges support from the Swiss National Science Foundation Postdoctoral Fellowships IMAGINE (No. 224226) and has received funding from the Research Foundation Flanders (FWO, G033625N). AT acknowledges the support by the Hasler Foundation grant Malescamo (No. 22050), and the Horizon Europe grant Automotif (No. 10114766 101147693).
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Appendix
Table of Contents
Extended background on causality
23
A.1 Pearl's framework of causality
23
A.2 Causal opacity
24
Concept discovery details
25
Causal graph discovery details
25
C.1
Causal discovery algorithms
25
C.2 LLM and RAG
26
C.3
Prompts
27
D
BMs detailed architecture
27
D.I
CBBM an universal approximator
29
D.2
CM interpretability
30
Dataset details
30
cMNIST
30
E.2
Bayesian networks
31
CelebA
31
E,4
Siim-pneumothora
32
E.5
CUB
32
Experimental details
32
Metrie: custom Structural Hamming Distance
33
Additional experiments
34
G.1
Label accuracy
34
G.2
Task accuracy using true graph
34
G.3
Ablation study on Causal Discovery
34
G.4
Ablation study on LLM type
35
G.5
Ablation study on RAG
36
G.6
Sample complexity of graph discovery
37
G.7
Sensitivity graph corruption
37
G.8
Effect of single-concept interventions
38
G.9
Decomposing interventional accuracy
38
A Extended background on causality
A.1 Pearl's framework of causality
Contemporary research in causal inference and causal machine learning predominantly builds on the framework introduced by Pearl (2009) (see also Pearl (2019). This framework centers on an agent's ability to reason about underlying causal mechanisms, going beyond mere statistical associations
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observed in data. Pearl formalizes this capacity through the notion of answering different types of what-if questions, structured into a three-level hierarchy (see Figure 6
COUTTEUCTIOALS
MASSINGG
INTERBNTION
Dopport
ASROGHY ON KETWTTA
CUTTOC:
Figure 6: Pearl's Hierarchy as represented in Pearl & Mackenzie 2018).
Level 1: Observational questions ("what is the value of Y if observe X = x?"".. These questions rely purely on statistical associations within the data. They can be answered using standard tools such conditional probabilities expectations, with need for causal assumptions.
Level 2: Interventional questions ("what is the value of Y if do X x?"). These questions focus on the effects of actively intervening variables, rather than passively observing them. The expression do(X = I> denotes such an intervention, where X is externaly set overriding its natural causes.
To compute such an intervention, one typically relies on a graphical model representing causal dependencies among variables, e.g., directed acyelic graph (DAG). The intervention is then modeled by removing all incoming edges X, effectively simulating the setting of X independently of its original causes. This modified graph is then used to compute the post-intervention distribution P(Y do x)).
Level 3: Counterfactual questions ("What would have been the value Y if had observed X = x' instead of X = x?"). These questions focus on states of affairs alternative to the actual reality. They constitute the upper layer of Pearl's hierarchy and their computation requires a detailed knowledge of the causal mechanisms relating the variables of the target- problem.
While standard machine learning models can generally handle only observational questions, reason- ing at interventional and counterfactual levels necessitates dedicated causal models and inference methods (Pearl) 2009 Peters al. 2017
A.2 Causal opacity
Problem A.1 (Causal Opacity). Given a DNN model M and an user A, we say that M is causally- opaque with respect to A whenever A is not capable to understand the inner causal structure of M' decision-making process. The opposite of causal opacity is causal transparency.
Note that causal transparency does not presuppose or imply causal reliability, the two issues, although related, are indeed very different. Consider for example the graph representing how the concepts are connected to the final task is shown in the left panel
sancept-based model Mr, where
"Thi excludes models specifically designed for causal inference, such as those in the field of causal machine learning (Kaddour et al. 2022)
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of the figure below (a). This model is trained to predict whether a given colored image represents an even odd number (variable P. for "parity") passing through a bottleneck of two interpretable concepts, namely "number" (N) and "color" (C). The structure of Mi 's decision-making process dausally-transparene including two edges connecting the final task P with N and C respectively. However, this causal structure is not consistent with the causal structure of the world (b), for which C' and P are clearly independent concepts Therefore, M₁ cannot be considered causally reliable, although it is causally transparent.
N
(a] Causal transparent graph
(b) Causal reliable graph
Causal opacity partially depends another opacity issue of central relevance for our analysis, i.e., semantic opacity.
Problem A.2 (Semantic Opacity). Given model M and a user A, we say that M semantically opaque to A if and only if MM' 's decision-making process is based on features that do not possess any interpretable meaning for A.
Semantic opacity is addressed by concept-based architectures, such as concept-bottleneck models and their extensions, which we discuss in the main paper.
B Concept discovery details
CBMs-like architectures tipically rely on labeled data for each concept, costly process that hinder their practical adoption. Label-free Concept Bottleneck Models (Label-free CBMs) (Oikarinen)
al. 2023) address this issue by automatically generating concepts and assign concept labels using pre-trained models.
In paper, apply approach similar to the one followed in Oikarinen et al. (2023) to the Piin-Pncumthorax dataset, which lacks concepts and concept annotations. Using GPT-4o, we first generate candidate concepts with specific prompt (see App. C.3). and apply multi-stage filtering process:
discard too long concepts (>50 character);
the CXR-CLIP model - a CLIP-based model pretrained on medical imaging datascts (Johnson ct al. 2019 Irvin et al. 2019; Wang al. 2017: You et al. 2023) to encode both concepts and class labels, and discard any concept with cosine similarity > 0.9 to either a class label another concept;
filter out concepts too similar to class labels to each other. Specifically,
discard concepts that are not sufficiently present in the training data. To this end, we also compute image embeddings using CXR-CLIP, and remove concepts whose maximum cosine similarity with images in the training set is < 0.2.
We then annotate images by computing their similarity to each remaining concept using CXR-CLIP embeddings, and binarize the resulting scores via 2-means clustering.
C Causal graph discovery details
C.1 Causal discovery algorithms
Algorithms for addressing the causal discovery problem be broadly classified into two main categories (Peters et al. 2017
Herre we restrict our discussion to causal discovery algorithms that assume the set of observed variables sufficiently captures the relevant causal influences. For more complex scenarios, e.g., (Peters et al. 2017
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Independence- based methods. These methods assume a correspondence between condi- tional independence in the data and graphical separation among variables, leveraging this relationship to infer the underlying graph structure. Typically, they recover class of DAGs that are equivalent with respect to conditional independencies, which can be compactly represented by CPDAG.
Score-based methods. These methods define a scoring function over potential graph structures and search for the graph that maximizes often using criteria such the Bayesian Information Criterion (BIC) (Peters et al.] 2017). The results from score-based methods often comparable to those from dnlependence-based approaches, as graphs that violate conditional independencies tend to result in poor model fits.
For our experiments, we adopted the Greedy Equivalence Search (GES) algorithm CCickering 2002 a score-based method that performed well in our setting, as shown in Table Bi in App. G.3
GES is a two-phase greedy algorithm that searches over equivalence classes of DAGs (CPDAGs). It begins with the empty graph and iteratively adds edges that yield the greatest improvement in the scoring function. Once local optimum is reached where addition improves the score the algorithm enters a backward phase, greedily removing edges that most increase the score. The core idea navigate the space of CPDAGs through local transformations (edge additions and deletions), using the score as a guide to optimize structure learning.
In our implementation, we the GES algorithm provided by the causal-leary Python li- brary (Zheng et al.) (2024), employing the Bayesian Dirichler equivalent uniform (BDeu) scoring criterion for discrete variables (Heckerman et al. 1995 Chickering 2002).
C.2 LLM and RAG
LLMs (Brown et al. 2020: Jiang et al. 2024) are capable of answering complex queries without
additional training. However, they can unreliable and prone to hallucinations (Ji et al., 2023] Zhang et al. 2023). y. Retrieval Augmented Generation (RAG) (Lewis et al.) (020)) addresses this by retrieving relevant textual information and appending it to the query, thereby improving the accuracy and reliability of LLM outputs. In the context of structural learning, LLMs have been utilized to construct causal graphs by employing ad-hoc prompts specifically designed to condition the model for answering causal queries (Antonucci al., 2023 Long al. 2023; Zhang et al.) 2024). To enhance the causal graph discovery process, we utilize an LLM integrated with a RAG to either direct or eliminate edges that remain undirected by the causal discovery algorithm. The information retrieval process is outlined as follows:
1. Document Retrieval: Given a causal query (e.g.. "Is lung cancer influenced by smoking?"), we first retrieve a set of documents from the web. In our experiments, we employed both the DuckDuckGo search engine for web pages and Arxiv for relevant paper abstracts. From each source, we retrieve the top 10 documents. For certain datasets (cMNIST, CelebA, and Sachs), local documents were used due to the unavailability or inaccessibility of information online (e.g., the Sachs paper). Each retrieved document is then segmented into smaller pieces, referred to as chunks, using a sliding window approach that samples 512-token chunk every 128 tokens.
2. Ranking: The causal query is transformed by the LLM using a query transformation approach (Gao et al.] 2023). which improves the semantic alignment of the query with the relevant document chunks. After transformation, all the retrieved chunks and the modified causal query are processed by a sentence transformer (Reimers 2019). specifically the www.tti--a-mpne--beeee baseet---a------- model. At this stage, cosine similarity is computed between the embedded transformed causal query and each embedded chunk.
3. Context: The LLM is then tasked with answering the causal query using the additional context retrieved in the previous steps. This context is derived from the top 5 chunks with the highest cosine similarity to the transformed causal query, providing the LLM with relevant and supportive information for generating more accurate answer.
All the prompts mentioned in this section can be found in App.
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Your query will be used to search scientific articles from the web. articles. C.3 Prompts
In this subsection we list all the prompts used in both the causal discovery part and label free concept generation.
DUCKDUCKGO SEARCH PROMPT
Your task in to create the nost affective search quary to find information that answers question.
Your query will be used to zearch the veb using web engine (e.g, google, duckduckgo). be
concise. This is the question: (question).
Provide the final query vithout brackets.
ARXIV SEARCH PROMPT
Your task is to create the nost affective search quary to find information that answers question.
given query, produce query help NOTE: be short and concise.
This is the question: (question). Provide the final query eithout brackets.
TRANSFORMATION QUERY PROMPT
Rephrase the query align senantically with sinilar target while maintaining its core neaning.
Output expanded query enclosed within
cexpanded_guery queryy tags (e.g. <expanded (exaiedd_eeryyreaeeee.eeeee (example_query) wttereuuurryrrrrreeeeeeeeee </espanded_ concise.
be very
Query: (quory) Expanded Query:
CAUSAL PROMPT
You are an expert in causal inference and logical analysis.
concepta you infer relationship *onceept 1:4* (concept_1) _1) (concept__1_ees d_descritio) (coacept_2) 2} {concept 2acapt____eee 2ddecciptio))))
provide you
e=Concept
Nov, your knoxledge and, available, the context provided, determine
which of the following options is the correct (A) changing [concept_1) 1) to certain valuea result in change in {concept. 2); changing
values result change (concept.1); (C) there is causal relationship or reciprocal influence betxeen (concept_1) and (concept_2).
The follouing information are extracted fron recent and reliable sources: {context)
The ansver haa to be enclosed within <ansver> tage (a.g. <ansver>\</nnvver>>)) step-by-step eneure final
Anslyze
CONCEPTS GENERATION PROMPT
You are expert of (context). list nost inportant List also the variables that are nost likely to be associated with [alass_label]] as sell
recognize (clase _lbbel) (input).
absence (class_label). You need also give list of superclasses for the vord (clase_label).
variablea that are nost likely to associated
Combine all the lists in single one and separate the single terus with a coma. underscore separate worda.
conposed by
D CBMss detailed architecture
In this appendix, we provide detailed description of the proposed C model training and functioning, using the Asia dataset as an illustrative example and Dyspnea as the task (Fig. 7). We assume the following information is available:
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A set of human-umaa understandable variables relevant to determining the task's value. Specifi- cally, the binary concepts: {Smoker, Bronchitis, Lung cancer, Either, Tubercolisis, Been in Asia, Xray anomalies, and Dyspnea).
A training dataset = (x,, V; V= where each sample is annotated with the values of all endogenous variables, i.e., the target variable Dyspnea and all preceding binary concepts.
A DAG G outlining the causal relationships between the concepts and the task.
Branchilis
Bronchitis
MLP
forome
P-
Task
Smoker
Dyspnea
Smoker
Dyspnea
MLP
f dapap
> P+.
p-
Lung cancer p
Lung cancer
MLP
Flung
Either
Either
MLP
Seith
Tuberculosis
MLP
frub
Tuberculosis
Been Asia
MLP
Been asia
Xray anomalies
Figure 7: Detailed C2BM architecture applied to the Asia dataset.
These elements can be either provided by human experts or generated using the pipeline we propose in the paper, described in Fig. 2 The proposed approach follows four main steps (Fig. 77. as detailed below:
Sub-causal Graph Selection. We extract from the DAG G only the variables that are ancestors of the task, that is, variables for which there exists causal path in G connecting the variable node to the task. In the case of the Asia dataset, the variable X-ray anomalies discarded because it is not an ancestor of the task.
2. Exogenous embeddings. Each variable (including the task) is assumed to have an associated embedding (the grey encoder symbols Fig. 7
latent factor, which represented learned from the input using dedicated neural encoder. In our implementation, these are implemented MLPs, preceded by -ptaset- feature extractor, e.g., a CNN for image data (as detailed in the App. EE).
3. Structural Equation Modeling, We model the structural relationships between parent nodes and their child nodes using linear equations. The weights of these equations predicted by separate hypernetworks, implemented as MI.Ps, which take as input the embedding of the child node produced in the previous step. Applying these functions normalized logits of a child node from the ones of its parents. For the nodes that lie in the roots of the causal graph (Been in Asia, Smoker), their logits are obtained directly from the corresponding exogenous embeddings.
derive the
For instance, we can calculate normalized logits for Dyspnea as follows:
Dayepeaa (0 10000ppm Bronckiits + 0pp
(8)
where PBronchitis and PEaher the normalized logits for the parents, θ fyyapnea ( 02) their corresponding weights and denotes a transformation function (such as softmax) applied to the weighted sum of the parent node logits, ensuring the final output in suitable range.
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While the structural equations are linear, the fact that the weights can be adaptively inferred from exogenous variables allows to capture complex dependencies between variables. This idea is analogous to locally approximating complex (smooth) functions. For example. consider an exponential relationship between two endogenous variables, V/2 = e (Fig. 8). This function can be locally approximated by a linear form V/2 = 01 V1, where the weight 01 is adjusted based on the value V1.
2/
V2 = Vn
V/2 = 1.8 Vi
V2 = 0.5 V1
0.3
0.8
V1
Figure 8: Non-linear functions can be modeled by adaptive re-parametrizable linear models.
is important to notice that such a model supports queries about any specific endogenous variable. Specifically, after training, C2BM can be used predict target variable (task), while using the other variables as concepts to explain the reasoning process. This provides greater flexibility compared to other concept-basee based architectures, which instead assume fixed task. Notably, only the task's ancestors are relevant in this case and, our implementation, all other concepts are discarded.
D.1
CM an universal approximator
We establish the following result regarding the expressivity of C.BM.
Theorem D.1. CM universal approximator regardless of the underlying causal graph.
Proof. We show that C BM can predict any endogenous variable as any DNN. Assume the exogenous encoder g(-) is a DNN (or any universal approximator) and that endogenous variables are represented as logits (the extension to fuzzy Boolean values is trivial). We consider four cases:
1. No endogenous parent: If V2 is root, V; MLP; (g()++) Since both g(+) and MLP; universal approximators, their composition is also a universal approximator.
2. Single endogenous root parent: If V; has exactly one root parent Vj, then V = , (x)g())))l VV, and since both r.) and g(-) are universal approximators, their composition is also a universal approximator. Multiplying this by V,. which itself is produced by universal approximator, preserves the ability approximate any function of the input.
3. Multiple endogenous root parents: V; has more than one parent, 0, can assign zero weights to all but one parent, reducing the case to single-paree parent scenario.
4. Non-root endogenous parents. The reasoning above can be applied recursively following the topological ordering of the causal graph. Fach variable computed as function of its parents, and universal approximation preserved layer by layer.
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Hence, by recursively composing universal approximators along the causal graph, CM can ap- proximate any mapping from the input endogenous variables. This holds for any graph structure, establishing CBBM as universal approximator.
D.2 BM interpretability
In this section, present explanation generated by C2 on the Asia dataset. As shown in Tab. [ the causal graph retrieved by the causal discovery mechanism is almost equal to the real one, despite missing edge between Been in Asia and libberculosis. Starting from the source endogenous variables, is possible to see the weight associated to each descending endogenous variable and the corresponding activation probability. For instance, Lung cancer is 'True' because the corresponding probability is peaked toward it (P(1) 0.99). In particular, the decision-making process for the classification of Dyspnea 'False' is completely unveiled. Although the parameter the edge from Bronchitis to Dyspnea promotes positive prediction, the stronger, negatively weighted connection from Either Dyspnea dominates. Resulting Dyspnea being predicted as 'False', worth noting that the endogenous variable X-ray anomalies is not considered by C'BM's inference since it an ancestor of the defined task.
Smoker
Been In Asia
True
x11
6.11
Tuberoulosis
True
True
4.96
p{1) 0.90
p(1) 0.99
4.38
2.79
Bronchilis False
True
0.99
p(0)
0.99
1.33
Dyspnea
X-Rey anomalles
Query
p(O)
Figure 9: Visualization example of information propagation (Asia dataset). The parent weight (predicted by the hypernetwork) visualized next to each edge.
E
Dataset details
E.1 eMNIST
The MNIST dataset (LeCun et al.) 2010) is large collection of freely available grayscale images of handwritten digits. It consists of 60000 training images and 10000 test images, both drown from the same distribution. Each image is labeled with the digit it represents. For our experiments, we download the original train and test dataset using the torchvision library (Marcel Rodriguez] (2000) and reserve 10% of the training set for validation. Additionally, we colorize each image based on its digit. Specifically, in the in-distribution setting (experiment Sec color is randomly assigned to each image in the training, validation, and test sets with equal probability over red, green, and blue. In the dut-of-distribution setting (experiment Sec 5.3), images the training and validation sets with odd digits are colored green, while the remaining images are colored red blue with
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equal probability. In the test set, images with even digits are colored green, and the remaining ones are colored red or blue with equal probability. For both the versions of this dataset, the following concepts are considered: Number, Color, and Parity (task). Finally, the images are preprocessed using a pre-trained ResNet-18 model with default weights from the torchvision library.
E.2 Bayesian networks
For our experiments, we use synthetic datasets sampled from discrete Bayesian networks available in the bnlearn repository (https://www. bulearn. com/ htmm//brepositor//// A Bayesian network is a probabilistic graphical model consisting of a DAG, where nodes correspond random variables, and each node is associated with conditional probability distribution (CPD). This CPD defines the probability of the node's value, given the values of its parent nodes in the network (Sharma et al. 2020). From the bnlearn repository, we select Bayesian networks with different dimensions and domains: Asia (Lauritzen & Spiegelhalter, 1988), a small network focused on lung disease with nodes and edges; Sachs (Sachs et al. (205), widely-used network modeling the relationships between protein and phospholipid expression levels in human cells with 11 nodes and 17 edges; Insurance (Binder et al., 1997). network for evaluating car insurance risks with 27 nodes and 52 edges; Alarm (Beinlich et al. 1989). a network designed to provide an alarm message system for patient monitoring with 37 nodes and 46 edges; Hailfinder (Abramson al.. 1996), network designed to forecast severe summer hail in northeastern Colorado with 56 nodes and 66 edges. For each network, we generate 10000 samples and create training, validation, and test datasets using a 70% - 10% - 20% split.
While node values can be used as concept annotations (v), input features (x) are absent. To generate them and make the datasets applicable to concept-based based architectures, we flatten the concept values and process them with simple autocncoder (MSE loss) comprising 2 encoder layers and 2 decoder layers (the latent dimension is adjusted based on the number of nodes: Asia-32, Sachs-32, Insurance- 32, Alarm-64, Hailfinder- - 128). Embeddings are further transformed that each sample is a mixture composed of 50% original data and 50% noise, with the noise drawn from a standard normal distribution. Finally, the output is standardized. The goal of these transformations is to make the inputs non-trivial representations of the concepts (the network nodes), forcing architectures to learn how to identify and retrieve the underlying concepts from the preprocessed data.
Modified versions of the Asia and Alarm datasets are considered, denoted as Asia" and Alarm", in which only subset of the original concepts is retained (experiment in Scc5.II. Specifically, for Asia", we keep only the concepts "Smoke" " and "Dyspnea". For Alarm", we retain only the concepts: "BP". "CO", "CATECHOL", "HR". "LVFAILURE", "STROKEVOLUME", "HYPOVOLEMIA".
E,3 CelebA
CelebA Liu et al.- 2015) is large-scale face attributes dataset with more than 200.000 celebrity images divided into training, validation and test set with 40 bi- nary attribute annotations. For our experiments, we first downloaded all the splits from the project website We then select subset of attributes that con- sider relevant for our analysis and apply this se- lection all the splits. These attributes are: At- tractive, Big Lips, Heavy Makeup, High Cheek- bones, Male, Mouth Slightly Open, Oval Face, Smiling, Wavy Hair, Wearing Lipstick (experi- ment in Sec P.Ip. For our fairness analysis (ex- periment in Sec 5.4). we selected the attributes: Attractive, Heavy Makeup, High Cheekbones, Male, Mouth Slightly Open, Oval Face, Pointy Nose, Smiling, Wavy Hair, Wearing Lipstick and Young. We then introduced two additional new attributes Qualified and Should be Hired. this analysis, we consider hypothetical scenario in which a person with a pointy nose is required for specific role, c.g., ... in movie. However, the
cuhk. edu. hk alk/projects/Celebe. html
Pointy nose
Heavy makeup
Wearing lipstick
Attractive
Qualified
Should be hired
Figure 10: Subset of introduced causal relation- ships among original and newly created concepts in CelebA, used for our faimess analysis.
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hiring manager has strong bias toward models who are considered attractive or heavily made up. Our aim intervene and mitigate such biases. The Qualified attribute is defined as a binary variable indicating whether a person meets the quali- fications for the job based on the hiring manager's biased criteria. It is therefore constructed using the logical expression: (Heavy Makeup and Wearing Lipstick) or Attractive. The Should be Hired attribute, on the other hand, indicates whether a person should be hired for the job, considering both the hiring manager's ' preferences and the role's requirements (having pointy nose). Therefore, it is defined as the logical "and" between Qualified and Pointy Nose. Additionally, we applied standard preprocessing to both versions of the dataset, including downsampling and normalization of the images, followed by feature extraction using a pre- pre-trained trained ResNet- 18 model.
E.4 Siim- poemmmneumttora
This dataset is derived from the publicly available chest radiograph dataset provided by the Na- tional Institutes of Health (NIH) and it contains chest x-ray images with binary annotations in- dicating the presence or the absence of Pneumothorax. For our experiments, we use in partic- ular the training annotations available on Kaggle (https://wwy kaggle cww/ comn//ommttiiiic o and the corresponding training images from https; 7/www. www.w..gge..... http///aaasttt/aaitteeeeeee /s/mmmimmmm.. png- images The dataset is then split into train- ing, validation, and test sets using traditional 70% 10% - 20% partition. As it does not include concept annotations, we followed procedure inspired by the methodology in inikariin al. (2023) to generate concepts and their corresponding annotations, More details can be found in App. B
8 siim-acr- pwtrrpppemmmmcooo---eeeeeeeeeeee
E.5 CUBC
This dataset is derived from the publicly available Caltech-UCSD Birds-200-2011 (CUB) (He & Peng 2019) dataset, which is widely used in the CBM community (Koh et al.) 2020 Zarlenga et al. 2022). It contains 11,788 images across 200 bird categories, with 5,994 images for training and 5,794 images for testing. Each image annotated detail, including 312 binary attributes. For our experiments, we downloaded the dataset from ht https: /data. cal tech. edu/recorr //doordd////d6--- (5de6-56666666 vp158 and further split the training set such that 10% is used for validation. We then selected the 112 most frequently activated binary attributes to serve as our concepts of interest, considered in Zarlenga et al. (2022).
To explore deeper causal relationships between concepts, we introduce four new ones: camouflage, flight adaptation, and hunting ability, which are derived from existing attributes using logical rules, as well as the multi-valued concept survival, whose activation in turn derived from these three binary concepts. The rules used for these derivations are detailed in the table below (Table We use survival as our downstream task. The images are instead further downsampled, normalized, and processed using ResNet-50 architecture, following a procedure similar to that used for the CelebA dataset.
F Experimental details
Python code and instructions for reproducing results across all datasets and methods are available within the code provided alongside the submission supplementary material. We detail below the configurations and hyperparameters used for models' instantiation and training. For a fair comparison across concept-based models, standardize the concept encoder single-hidden- layer MIP. All models are trained using the Adam optimizer (Kingma & Bal 2015) for a maximum of 500 epochs, with early stopping based on 30-epoch patience. We employ LeakyReLU as the activation function throughout.
The batch size is set to 512 for most datasets, with the exception of piim-nneuot and SCBM, where it is reduced to 128 due to memory constraints. Following Koh et al. (2020), we regularize the task loss encourage concept learning, using weighted sum of task and concept losses:
I. (1 a). Tank + a Tecuncepts: with a = 0.8
where Luck is a cross-entropy over the task whereas Leancepts is the summation of cross-entropy losses over the concepts.
Additionally, we apply random training-time interventions as proposed by Zarlenga et al. (2022). with intervention probability of 0.25. Regarding SCBM, used the authors' implementation
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Table 4: Newly introduced CUB concepts and the rules used determine their values.
New Concept
Logical rules
camouflage
has Laai__pateer__polleeeeee V
has tail. pattern pattem_striped has tail pattern multi-colored V
has back htt__aaaccctcaa____e_eee
cuuteem_spotted
has Balii__attt__mulliiooooto www.immclloreeee has mas_back BaiisK_patte______
plt__patttttttpeeeepe s_ttrrssssppeedte V
saii___aa__rrudddd_____e has has_tail_ sis_tall_shappe rounded _tall V
has sa__wing__saee wwwpsshhhaadseeeeeee rwwndodd-wnnnee wings
has has_size medium
sis_szze_medium
hunting ability
has bill cchape_curver V
has mas_bill___ sa___shaap__ee___ _eedle has_bill, whalbllshhpeesspatlltttt V
has gas_bill_shape. paapppp_allpuurposeeeeeee
has has_bill_shaae biill_shape_longe___
_ead V
has has_bill bill_shape_shhrtee hathape_shorterthaate tiaiier_than_haar
survival
max(camouflage + flight adgpttaianttionn hunt-
ing_ hig_ability, 2)
/ More precisely, implemented the global variation using the configuration proposed by the authors.
Key hyperparameters, including learning rate, MI.P hidden size, and dropout rate, are selected via grid search. complete list of hyperparameters for CBBM and all baseline models can be found in the provided YAML configuration files within the code.
All experiments are conducted on NVIDIA GeForce RTX 3080 and NVIDIA RTX A5000 GPUs.
F.1 Metric: custom Structural Hamming Distance
To assess the quality of the causal graphs generated by our pipeline and baseline discovery models, we employ two metries: variant of the structural Hamming distance (SHD that operates CPDAGs, and the number of incorrect edges identified. The number of incorrect edges serves as standard measure of the quality of the graph, while the SHD allows us customize weights for different types of errors. Tab. 5 provides schematic of our SHD scores:
Ground Truth
Predicted
SHD Penalty
/
j
1/2
+ j
i ++] j
1/3
i + J
1/4
1/4
i 1
1/4
i->
i-j
1/5
Table 5: SHD penalties for various discrepancies between ground truth and predicted edges. 'i + jr: oriented edge, j': unoriented edge, /': no edge
The rationale behind the scores is that the insertion of edges is 'riskier' than the removal of existing ones. The introduction non-existing edges may induce spurious correlations that strongly affect the reliability of the model and related metrics, such counterfactual fairness accuracy in ood tasks. On the contrary, removing an existing edge is more conservative operation: while it still
Although refer this "distance," it is technically asymmetric scoring function.
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affects the model accuracy, is not strongly impacting reliability. For the same reason, introducing an incorrect edge orientation more penalized than removing an orientation.
G
Additional experiments
G.1 Label accuracy
We report here the average label accuracy (task + concepts) for each of the datasets analyzed Tab. In this setting, since we are evaluating the accuracy across both the task and the concepts, we use a different opaque neural baseline, OpaqNN M. which jointly predicts concepts and the task for each dataset. As shown in the Tab, 6 C achieves comparable results to non-causal models in terms of concept prediction. Notably, the performance differences observed in Section 5.1 for task accuracy attenuated here due to averaging over both tasks and concepts.
Table 6: Label accuracy (%). Task concepts
follows: dysp (Asia), Akt (Sachs), BP (Alarm), PropCost (Insurance), R5Fest (Hailfinder), parity (cMNIST), mouth slightly open (CelebA), Survival (CUBc), pneumothorax (Pneumoth.). Uncertainties represent sample mean across 5 runs.
MODEL
SEMANTIC TRANSP:
CAUSAL
ASEA
SaCHS
INSURANCE
ALARM
HAILINDER
CMNIST
CELERA
CUBc
PNEUMOTTH.
REL.
OragNNN
x
87.1an
72/4g1.
78. 73.7ian-m
900.13gn.as
06.29gn
91.822
80.80gm
17.06g..
74.99gn
CBMati
87.0gx4
71.9
78.60gi-s
60.99gaa7
91.83
79.98
77.58gs
87.1asa
72:243.a
80.80
60 78.78
91.48..u
80.05
77.7540.51
70.72
CFM
87.1188
72.213.3
78.0010.7
80.77.6.2
70.14
91.421.77
80.59.e:2
77.16.6.31
74.271077
87.0gna
72:2g3.
m
90.08
91.84.
7U.932e:3
78.00
74.3120
C'BM
87.2g11
72.0gn
78.37
89.83
71,80go
92.19;
80.44go
77.22gr
74.78kma
G.2
Task accuracy using true graph
To further assess the quality of the generated graph, we compare downstream task performance using the inferred graph with performance using the true graph, available in Bayesian Network synthetic datasets.
Table 7: Task accuracy (%) using the true and predicted graph. Task concepts are follows: dysp (Asia), Akt (Sachs), PropCost (Insurance), BP (Alarm), R5Fest (Hailfinder). Uncertainties represent 2 sample mean across 5 runs.
MODEL
ASIA
SACHS
INSURANCE
ALARM
HAILFINDER
C²BM
TRUE GRAPH
71.0+2.3
65.0
66.4
62.1
73.0
PREDICTED GRAPH
71.4
65.3
66.4
62.5
74.1
Results in Tab. 7 iidicatee that the two settings yield comparable performance, demonstrating both the quality of the learned graph and the robustness of the proposed pipeline. This robustness can be attributed to the employed exogenous latent embeddings, which mitigate the impact of concept incompleteness. Consequently, even when the inferred graph is not perfectly aligned with the true causal structure, BM maintains strong task performance. These findings highlight the model's resilience and its ability to generalize effectively in real-world scenarios where true causal structures are often unavailable.
G.3 Allaioon study on Causal Discovery
In this section, we evaluate the sensitivity of the causal discovery component in our pipeline to the choice method for constructing a causal graph. do so, we compare our method choice for causal discovery, i.e., Greedy Equivalence Search (GES) (Chickering) 2002), with different widely-used causal discovery algorithms that recover a CPDAG. Each of these methods evaluated with and without refinement via the retrieval-augnee augmented generation (RAG) language model
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(LLM) employed in our study, Furthermore, we compare all these methods against the use of the LLM alone, as well as our retrieval-augee retrieval-augmented LLM (LLM RAG) applied directly to discover the whole graph. Specifically, the LLM is prompted with the causal prompt shown in App. C.3 with additional retrieved context appended to the query in the L.LM+RAG setting.
Specifically, we evaluate the following methods from the causal discovery literature:
GES (score-based): The algorithm selected in our study. A detailed description is provided in App.
PC Algorithm (Spirees et al. 2000) (independence-basel) A classical independence-basee method that first estimates the undirected structure the causal relations through sequence of conditional independence tests, then orients edges using set of predefnnd orientation rules (Colombo et 2014). our implementation, we use the PC algorithm provided by the causal-learn library (Zheng et al. 2024). with a chi-squared independence test and
significance level of 0.05;
Fast GES (Ramsey et al. 2017) (FGES) basoee-based): computationally efficient variant of GES that improves performance by storing intermediate evaluations and parallelizing expensive operations, enabling its application large datasets (Andrews al. 2019 Ramsey et al. (017)) In our implementation, we use the FGES algorithm provided by the py tetrra library (Ramsey & Andrews 2023), employing the Bayesian Dirichlet Equivalent Uniform (BDeu) score for structure evaluation (Heckerman et al. 1995).
The results are evaluated on all the bnlearn datasets, for which the true causal graphs are known. Results are presented in Tab. S
Table Structural Hamming distance and number of mistaken edges between the true and learned causal graphs for each tested method. The maximum number of errors (Max errors) and the average number errors make by random classifier (Random Classifier) are also provided for reference.
Metric
CD Method
Asia
Sachs
Insurance
Alarm
Hailfinder
Hamming
LLM
11.08
336
505.92
92.75
LLM RAG
12
6.83
201.91
Time limit
Time limit
PC
2.41
2.93
6.65
14.53
PGES
0.65
3.4
6.78
5.91
21.4
GES
0.65
3.4
6.43
5.41
PC + LLM (+ RAG)
2.41
2.91
6
3,92
15.42
FGES + LLM (+ RAG)
0.25
2.03
5,33
20.5
GES + LLM (- RAG)
0.25
1.83
6.33
4.95
11
Number misraken odges
errors Random Classifier
55
18.33
117
222
513.33
20
543
LLM RAG
12
226
Time limit
Time limit
10
FGES
17
GES
17
19
PC + LLM RAG)
9
FGES + +LLM (+ RAG)
8
15
GES + L.M (+ RAG)
7
18
As shown in the results, GES refined with the RAG-augmented LLM is the overall best-performing method in the majority of cases, both in terms of structural Hamming distance and number of mistaken edges. In instances where it is not the best, it still consistently ranks among the top methods. Notably, both the standard causal discovery method (GES) and the use of the RAG-auumented LLM contribute positively to performance. Due to the substantial computational time required by the HLL+RAG--basee causal discovery approach, we excluded experiments on datasets for which execution exceeded 24 hours.
G.4 Ablation study on LLM type
The LLM or the design of the L.MM prompt could condition the causal graph refinement. To assess this, present ablation study in which we fix the incomplete causal graph generated by the causal
35

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discovery algorithm (GES), and later evaluate different LLMs, with different prompts, for the graph refinement step. Specifically, we evaluate the impact of four different prompting strategies with three I.MMs (GPT-40, 200M parameters; and GPT-40-mini, 8M parameters). The considered strategies are as follows:
Minimal prompting: simply asks the LLM to identify causal relations without additional guidance;
Instruction prompting: provides a more detailed explanation of what constitutes causal relation;
Few-shot prompting: proposed in Ye et al. 2024). combines instruction prompting with few examples;
Chain-of- (CoT): proposed in Wei et al. (2022 encourages the model generate
intermediate logical steps before generating the answer. This is the approach we used in the original manuscript.
Tab. compares the number of mistaken edges and the custom Hamming distance between the true and the learned DAG. Causal reliability of standard, flat, CBMs also reported for reference. The results show that, using GPT-5, causal reliability improves on average w.r.t. GPT-4o, and GPT-4o-mini, especially on the Sachs dataset. Despite moderate variation the Sachs dataset, performance is generally very robust to the prompt strategy (made an exception for naive minimal strategy), and causal reliability is largely superior to standard CBMs all cases.
Table 9: Structural Hamming distance and number of Mistaken Edges between true and learned DAG across datasets, using different LLMs and prompting strategies. Uncertainty represents sample mean Ø across runs.
Metric
L.M
Prompting Strategy
Asia
Sachs
Insurance
Alarm
Hailinddr
Minimal
0.25:1.00
2.67:1.0
8.04-..22
3.631.94
10.83
Instruction
0.25
2.75:1006
8.04
4.3
10.83
GPI-4o-mini
Few-shot
0.25
2.79
8.04
4,3411.2
10.83-:51
CoT
0.252:00
2.21
7.672
4.12:111
10.83
Minimal
0.25+:.0
3.754.98
Hamming
GPT-4o
Instruction
0.25:::00
2.63 16
8.04
4,7+13
10.83
Fea-shor
0.25:1:00
2.46:1.0
8.04
4.213
10.85
CoT
0.25
2.25
7.54
4.7
10.83=:51
Minimal
0.25++.00
1.294:08
Instruction
0.25:t:00
1,21+
7.29
4.4+112
10.83
GPT:5
Few-shor
0.25:.00
1.04
7.29::44
4.0+1.11
10.85
CoT
0.25:1.00
0.92:1.9
7.172.39
4,631
10.83:1511
Minimal
1000:.00
23.0
9.0
21.50=1.90
GPT-4o-mint
Instruction
1000::00
9.50
23.0
10.0+222
21.50=:90
Few-shot
1.00
9.50
23.0
1000:2
21.50=990
CoT
1.00::.00
8.00::336
21.51:3
9,0
21.50::.90
Mistaken Edges
Minimal
1.00
11.00
23.0
9.5
21.50=290
Instruction
1.00.1.00
10.00:: 72
23.0
10.0:12.2
21 50 90
GPT-4o
Fow-shet
1.0.1.00
21.50:1.90
CoT
1.00::.00
8.50
21.021.4
10.0:12.2
21.50=290
Minimal
1.00::000
4.50-2.18
22.5+113
9.5
21.50=:90
Instruction
1.00.1.00 00
4.00:: 00
20.0
9.0
21 10.50=
GPT-5
Few-shot
1.00.1.00
3.50.1.18
20.0:1.11
9,0
21.50-2.90
Col
1.00+:00
3.00
19.5
9.5+22.0
21.50=:90
G.5 Ablation study on RAG
Tab. 10 illustrates the influence of the context supplied by RAG in correcting the causal graph gener- ated by the causal discovery algorithm. In this analysis, we used GPT-4o for both the experiments with and without RAG, aiming evaluate the effect of context answering causal queries.
Although the context provided by RAG appears to have no significant impact on final causal graph for the Alarm dataset, it proves essential for correctly handling the undirected edges in the causal graph for the Sachs dataset. We hypothesize that this due to the absence of protein related documents used in training the LLM, which leaves it with insufficient prior knowledge to address
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Table 10: Structural Hamming distance and number of mistaken edges between true and learned DAG when using either RAG to provide context the LLM just the LLM.
Metric
Context
Sachs
Alarm
Hamming
No context
3.1
RAG context
1.8
5.0
Mistaken
No context
12
10
edges ratio
RAG context
7
10
specific questions on the topic. RAG helps mitigate this limitation by supplying the LLM with the relevant information, thereby compensating for the lack of prior knowledge. In conclusion, while an LLM with the ability to fully comprehend complex queries is crucial for causal discovery, the additional context provided by RAG is vital overcoming the LLM's prior knowledge gaps.
G.6 Sample complexity of graph discovery
We study the effect of dataset size and graph size on the quality of CBBM's graph construction pipeline. We explored this with sensitivity study, running the causal graph pipeline (causal discovery with GES + LLM refinement with CoT prompt) across all datasets with an available ground-truth graph and comparing the number of mistaken edges between the and the learned DAG. For each dataset, we varied the number of data points N from 100 to 10000. Results are presented in Tab.
Table 11: Number of mistaken edges between the true DAG and the learned graph using the C2BM's graph construction pipeline, evaluated across datasets and data sizes. For reference, the reliability of standard flat CBMs is also reported.
Dataset
Flat CBMs
B2BM's causal graph
Data size (N)
100
500
1000
5000
10000 (Paper)
Asia
11
3
1
Sachs
15
12
11
7
7
Insurance
74
47
35
24
22
18
Alarm
78
31
16
9
9
9
Hailfinder
117
56
47
44
22
22
few considerations emerge:
C2B''s causal graph is consistently more causally reliable than the flat structure implicitly assumed by standard CBMs, regardless of the dataset size.
As expected, increasing the number of data points leads to better alignment between the estimated and true causal graphs. Causal reliability tends remain stable larger data sizes.
We observed that the data size threshold for reliable performance does not strictly depend on graph size. We speculate this is due to the varying impact of LLM-based refinement across datasets. In some cases, an effective background knowledge can compensate for limited data.
G.7 Sensitivity to graph corruption
In this section, we empirically assess the robustness of C2BM to graph misspecification and corruption. Although CM is theoretically universal approximator for the final prediction task, independent of the specific causal graph (see Appendix D.II). complement this result with empirical validation.
Adversarial Graph Corruptions. We first evaluate robustness by altering percentage p of graph edges, chosen randomly, with one of the following operations: edge flipping, addition,
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or removal. The resulting performance across datasets and corruption levels is reported in
Tab. 12
Progressive Flattening into Standard CBMs. As a second evaluation, we progressively transform the graph into the flat structure assumed by standard CBMs, by connecting percentage of nodes directly to the prediction task while removing their outgoing edges. Results shown in Tab. 13
Table 12: Task accuracy (%) under edge-level adversarial corruptions. A percentage of edges is altered through flipping, addition, or removal. Task concepts follows: dysp (Asia), Akt (Sachs), PropCost (Insurance). BP (Alarm), R5Fest (Hailfinder). Uncertainties represent sample mean o across 3 runs.
Dataset
0.05
0.1
0.2
0.4
0.6
0.8
1.0
Asia
71.840.6
71.6+0.5
71.40..6
71.7+1.3
70.2+1.9
7..41.2
70.2
Sachs
65.1
65.6
64.942.0
64.6+1.2
65.5
65.2
65.042.0
Insurance
67.2.t..6
67.043.11
67.5
66.1+1.1
67.0.2..6
66.8
67.2
Alarm
61.94:2.6
61.7++2.1
61.5-+2.5
61.9+2.7
60.6+1199
60.9
61.44+1.7
Hailfinder
74.0
73.1
72.342.2
72.313.2
73.042.1
71.841.6
72.411.4
Table 13: Task accuracy (%) under progressive graph flattening into standard CBMs. A percentage p of nodes is directly connected to the task output. Task concepts are as follows: dysp (Asia), Akt (Sachs), PropCost (Insurance), BP (Alarm), R5Fest (Hailfinder). Uncertainties represent sample mean σ across runs.
Dataset p
0.05
0.1
0.2
0.8
1.0
Asia
72.0+1.00
72.0+1.0
7..5
71.7+0.5
71.7+0.55
7..2+
71.111.1
Sachs
65.5++2.2
64.8+3.0
65.8++1.
64.7+1..
65.2+1.2
65.1+2.7
64.9+2.2
Insurance
67.412.6
66.2
66.342.6
65.643.3
65.3
66.941.8
64.743.1
Alarm
62.642.9
61.612.6
61.642.00
60.5
61.211-8
60.6133.2
61.341.9
Hailfinder 13.1
73.2.1.9
72.5
72.712.9
73.042.6
73.0
72.941.9
Across both corruption strategies, we find that task accuracy remains stable, even when the causal graph is heavily perturbed. These empirical results support the theoretical claim that C2BM is robust to graph misspecification.
G.8 Effect of single-concept interventions
Fig. presents the relative improvement in task accuracy after intervening on individual concepts across all datasets. A key observation is that CMBM responds to interventions on the same key concepts as the baselines, despite the fundamental difference how information propagates. In CBM-based models and CEM, all concepts are directly connected to the task, enabling direct influence. contrast, C²BM enforces information flow through the causal graph, constraining the interactions. Yet, the task performance improvements remain consistent across models. These results highlight that CB preserves the intervention effects observed in traditional concept-based models while providing more structured and interpretable causal representation of the underlying relationships.
G.9 Decomposing interventional accuracy
Fig. in the main paper illustrates the improvement in cumulative relative interventional accuracy across all downstream, non-intervened concepts, including both intermediate concepts and the final task. To further analyze these effects, Fig. decompose this metric into two separate evaluations: one focusing solely on the task node and another considering only intermediate concepts. The concept
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Asia
Sachs
Rel. Improv. (56 on task accuracy
Dall imganx (89) on tock arsuacy
-CBM.e SCBM CIBM
20
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Adhing
1114.
brone
ether
lung
smoke
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xay
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Mek P38 PIP2 PIP3 PKA PKC Plog Raf Intervened concept
intervened concest
Imprrv. (96) accuracy
Insurance
Alarm
8
impiox on
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PRO
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Hailfinder
Rel. Improv. (9) on task spcurE
CelebA
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Rel. improv. (96) accuracy
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Figure 11: Relative improvement (%) in task accuracy when intervening specific concepts.
accuracy plots highlight C²BM's unique ability to enhance performance of intermediate (downstream) concepts, a property observed in competing models.
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06
Asia
85
Asia
100
Asia
Label 600 acouraby (90)
14
accuracy
82
U
Concept
88
Task
70
40
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86
Number intervened concepts
Number intervened concepts
Number concepts
Sachs
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100
Sachs
Concept accuraty (34)
95
Label accuragy (96)
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U
60
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1
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Number of intervened concepts
Number of intervened concepts
Number of intervened concepts
100
Insurance
72
Insurance
Insurance
Label accuracy (96)
700
Concept (96)
A5
Task accuracy
66
80
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Alarm
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Labblt accuraiy (99)
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E77
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U
96
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86
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Hailfinder
100
Hailfinder
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Hailfidder
Label accuracy (90)
8
Task Task accuracy (96)
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a
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5 10 15 20
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30 35 4n
D. 5. 10 15. 20 25 30 38 40 45
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40

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CelebA
CelebA
100
CelebA
Label acouracy (90)
Task aecuraco (90
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U
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Concept
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70
Number of intervened concepts
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CUB
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Label accuracy (96)
28
Task accuracy (56)
95
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Number intervened concepts
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Pneumothorax
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Pneumothorax
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06
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Label accuracy (96)
96
Task accurac
410
80
86
28
U
76
74
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3 Number of intervened concepts
3
Number of intervened concepts
Number of intervened concepts
CBM+lin
CBM-mip - CEM
SCBM
C2BM
Figure 12: Task accuracy improvement (%) in predicting downstream variables after intervening on groups of concepts up to progressively deeper levels in the graph hierarchy. The metric is averaged all downstream variables (Left), the task only (Middle), and all concepts (Right) Total label accuracy. Uncertainties represent 2 sample mean or across runs.
41


Reviews:
[{"review_id": "NJGgE8jnM5", "reviewer": "Reviewer_wNgz", "summary": "This paper proposes Causally reliable Concept Bottleneck Models (C2BMs), which explicitly incorporate causal structure into the concept bottleneck architecture. The authors present a novel pipeline to automatically learn the causal concept structure from observational data and unstructured knowledge sources, aiming to enhance interpretability, causal reliability, and responsiveness to interventions.", "questions": "1. The notion of a \"bottleneck\" presented in the paper is intriguing. However, the manuscript lacks details on how the bottleneck is quantitatively evaluated and how this mechanism influences the model’s operations. I recommend the authors provide a more thorough explanation and possibly include experimental evidence demonstrating its effect.\n\n2. Section 4 cites and utilizes many existing methods, which may obscure the unique contributions of the current study. To improve clarity, the authors should explicitly state the advantages of their approach and clearly differentiate it from prior works.\n\n3. The proposed model does not appear to achieve significantly better task accuracy compared to baseline methods. The authors should discuss possible reasons for this and highlight the practical or theoretical benefits that justify their approach despite the modest performance gains.\n\n4. The training procedure and other experimental details are insufficiently described. Please provide a comprehensive explanation of how the model is trained. Additionally, we suggest including ablation studies from various perspectives discussed in Section 4, such as concept discovery, labeling, and other relevant components, to better understand the contribution of each part.\n5. As the authors mentioned, a robust prior knowledge base is very important to model training and learning. This part is interesting, but not including many details. Suggest the authors explain how to built such a knowledge base. and answer how to evaluate the effectiveness of the built knowledge base, cause that can impact the training performance. Is the pipeline for constructing the knowledge base fully automated? How is the transferability of the knowledge ensured?", "limitations": "yes", "rating": 2, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "The research tackles an important challenge at the intersection of explainability and causality in deep learning and offers promising directions. However, further clarification on the automatic structure learning process and more extensive evaluation would strengthen the manuscript.", "was_revised": true, "final_justification": "I have carefully read the author's response and, after rigorous evaluation, I believe that this paper does not meet the acceptance level of NISP. I retain my previous rating.", "rebuttal": "We thank the reviewer for their comments and feedback. As for the raised concerns and questions, individual points are addressed below.\n\n***Q1 - Bottleneck evaluation***\n> the manuscript lacks details on how the bottleneck is quantitatively evaluated and how this mechanism influences the model’s operations.\n\nThe standards of evaluation of concept bottlenecks [1] are well established in the CBM literature [1,2,3,4,5,6]. These are evaluated in terms of **interpretability** (e.g., concept accuracy) and **actionability** (e.g., interventions) (ll.19-26).\nMatching task accuracy of black-box models is also an important desideratum (ll.70-72), yet not trivial when conditioned to provide human-interpretable and actionable models. In fact, this often exceeds typical expectations of the CBM community [7].\n\n\nRegarding our proposed *causal* concept bottleneck, **we disagree that a quantitative evaluation is lacking** in our manuscript. Section 5 provides extensive evaluations of our causal bottleneck, assessing standard CBMs' desiderata (see above) while also evaluating further specific benefits of our model. \n\nSpecifically:\n- (Task acc.) **Tab. 1** shows that our causal bottleneck preserves the task accuracy of black-box models, in line with CBMs' desiderata.\n- (Concept acc.) **Tab. 6** shows that our causal bottleneck preserves the concept accuracy of prior CBMs.\n- (Interpretability) **Tab. 2** shows our bottleneck provides interpretability at a deeper level by aligning more closely with ground-truth causal structures than prior CBMs.\n- (Actionability) **Fig. 4** shows how our bottleneck permits more effective interventions than prior CBMs. **Fig. 5** and **Tab. 3** evaluate two C$^2$BM's critical benefits: permit human interventions to de-bias the model from spurious correlations and unfair behavior.\n\nWe hope this clarifies that our experimental evaluation is centered on the causal bottleneck and its contributions.\n\n\n***Q2 - Clarify distinction with prior works***\n> Section 4 cites and utilizes many existing methods, which may obscure the unique contributions of the current study.\n\nThank you for your constructive comment, as it allows us to improve clarity.\n\nOur core contribution is the C$^2$BM model (right side of Fig. 2), a novel extension of CBMs that, instead of assuming a flat layer of independent concepts,  forces inference to flow through a **given** causal graph of concepts (ll.73-76). As stated in the paper (ll.42-44, ll.140-141, Remark 4.1 in ll.150-152), C$^2$BM requires **as input** a set of concepts (as most CBMs) and a causal graph connecting them; these are **distinct** elements that must be provided prior to C$^2$BM training. \n\nHowever, these requirements are often not readily available in real-world scenarios (ll.140-142). We argue that fully automating the instantiation of C$^2$BM is both valuable and non-trivial. For this reason, we present a pipeline that enables full automation (ll.142-145) as a second contribution of our work; this justifies its placement in the Method section. While the individual components of the pipeline are mostly based on prior work (hence the citations), their integration into a coherent, automated process -- which was never published before -- is novel by definition.\n\nTo improve clarity, we have added a clarification in the main text after l.149: \"In the following, we outline our implementation for each sub-problem. Specifically, building C$^2$BM's individual prerequisites will be mostly based on prior work. Note that integrating them into a coherent, automated pipeline is instead part of this paper's contributions.\". We thank the reviewer for raising this point and hope this resolves the concern.\n\n\n> the authors should explicitly state the advantages of their approach and clearly differentiate it from prior works.\n\nC$^2$BM is a kind of CBM architecture, **not** a method for concept or causal discovery, hence these do not constitute direct prior work from which C$^2$BM needs to be differentiated. In the paper, we explicitly state our advantages in terms of actionability and causal reliability w.r.t. prior methods in the CBMs community: in the introduction (ll.27-37), preliminaries (ll.73-76), related works (ll.110-112), and conclusions (ll.351-355).\n\n\n***Q3 - Task accuracy***\n> The proposed model does not appear to achieve significantly better task accuracy compared to baseline methods.\n\n**We firmly disagree that this should be seen as a weakness.** The ultimate goal of our field of research is to improve on interpretability and actionability aspects while **preserving** the task accuracy of traditional black-box approaches [2,6]. As demonstrated in Tab.1, C$^2$BM matches the performance of the corresponding black-box architecture, which is considered an upper limit in the interpretability community and generally difficult to achieve [7]. Furthermore, we support this result with a proof stating C$^2$BM is a universal approximator for the downstream task (see Appendix D.1). Therefore, the rest of the experimental analysis focuses on aspects critical to the interpretability community, i.e., reliability, actionability, and fairness. \n\n\n***Q4.1 - Training details***\n> The training procedure and other experimental details are insufficiently described.\n\nWe disagree. C$^2$BM's training objective is formally described in **Eq.6**, i.e., maximum likelihood (similar to other CBM classifiers [2]) conditioned on endogenous parents' values and meta-model output. Full experimental details, including numerical values for training hyperparameters, are provided in **Appendix~F** (referenced in ll.243-244). Finally, for the most comprehensive list, we provide the **code** with all settings specified in the YAML configuration files. \n\n\n***Q4.2 - Ablations***\n> we suggest including ablation studies from various perspectives ... such as concept discovery, labeling, and other relevant components\n\nWe value the reviewer's suggestion. A few ablations were already included in Appendix G due to space constraints. Specifically: (i) **Appendix G.4** provides an extensive ablation on the causal discovery block. (ii) **Appendices G.5–G.6** analyze components related to querying the LLM. In addition, we have now included: (iii) a new ablation on the LLM prompt (please refer to our response **W1 to Reviewer 6yXN**, and (iv) a study on robustness to corrupted causal graphs (response **W2 to Reviewer NX3e**).\n\n\n\n***Q5 - Background knowledge***\n> explain how to build such a knowledge base. ... Is the pipeline for constructing the knowledge base fully automated?\n\nThere might be a misunderstanding here. **We do not build the knowledge base ourselves**. Instead, as in ll.180-181 and detailed in Appendix C.2, we leverage a **pretrained LLM**, which implicitly encodes broad, **unstructured** knowledge acquired during its training phase [3]. We further improve this with a RAG strategy, which allows the model to take archived domain-specific knowledge (e.g., scientific papers), available through public sources such as *arXiv*, automatically into account (ll.97-99 and Appendix C.2).\n\n\n> how to evaluate the effectiveness of the built knowledge base\n\nWe evaluate its effectiveness by measuring how much it improves the alignment between the learned graph and the ground-truth graph (see '+LLM' row in **Tab. 2**). This structure constrains C$^2$BM reasoning, making predictions and explanations intrinsically more aligned with real-world causal mechanisms.\n\n\n> How is the transferability of the knowledge ensured?\n\nTransferability of the knowledge base to the model is ensured **by construction**: each concept value is predicted from its causal parents as defined in the graph refined with the knowledge base. This forces a principled flow of information aligned with expert knowledge. Nevertheless, we are not sure we fully understood the reviewer's question. We kindly ask the reviewer to reformulate should our response be off-target. \n\n\nWe hope our responses offer a clearer perspective on the paper and help position its contribution within the interpretability community. We are happy to further engage in the discussion.\n\n**Refs:**  \n[1] Koh et al., Concept bottleneck models, ICML, 2020  \n[2] Zarlenga et al., Concept embedding models, NeurIPS, 2022  \n[3] Oikarinen et al., Label-Free Concept Bottleneck Models, ICLR, 2023  \n[4] Zarlenga et al., Learning to Receive Help: Intervention-Aware Concept Embedding Models, NeurIPS, 2023  \n[5] Poeta et al., Concept-based explainable artificial intelligence: A survey, Arxiv, 2023  \n[6] Ismail et al., Concept Bottleneck Generative Model, ICLR, 2024  \n[7] Arrieta et al., Explainable Artificial Intelligence (XAI): Concepts, taxonomies, opportunities and challenges toward responsible AI. Information fusion, 2020"}, {"review_id": "cb1SfrZ2Ws", "reviewer": "Reviewer_6yXN", "summary": "This work deals with the task of building interpretable predictive models for raw data like image and text. While existing methods like Concept Bottleneck Models (CBMs) are transparent in their decision-making, their explanations are not causal, that is, they cannot be used to recommend interventions. This work extends CBMs by making them causal, calling the result Causally reliable Concept Bottleneck Models (C2BMs). This model consists of a set of concepts, a structural causal model (SCM) over the concepts, a neural encoder mapping raw data to a set of latent representations, and a neural encoder mapping each latent representation to a set of weights, used to define structural equations of the SCM.\nThe authors do so using concept bottleneck, that is, using a neural network to map the raw data to a set of interpretable concepts, which are used to make the final decision. The authors extend CBMs by combining them with causal models, learning neural encoders to map raw data to structural causal models of concepts, allowing reliable causal inference.\nThe proposed method consists of three stages. The first stage is concept discovery; they use label free concept discovery and then filter the concepts based on brevity, distinctness and presence in their training data. A pretrained vison-language model like CLIP is used to annotate each data point with these concepts. The second stage is causal discovery; they address this by first constructing a CPDAG using greedy equivalence search and then orienting its edges by querying an LLM. To improve reliability, retrieval augmented generation (RAG) and majority voting across 10 repetitions is used. Finally, the third stage involves learning the structural equations; they approximate each such equation for each variable as a weighted sum of its parents’ values, each parent weighed by the neural encoder’s output. Each structural equation is learned separately, exploiting the Markov condition due to the SCM’s causal graph.\nThe authors evaluate this approach on 5 BN-based domains, 1 benchmark domain color MNIST (cMNIST) domain, and 3 real-world domains: CelebA, facial recognition, annotated with binary facial attributes; CUB_c, a subset of original CUB birds data set, having a subset of concepts and some new ones to introduce deeper causal relationships; and Siim-Pneumothorax, consisting of chest radiograms, annotated with a single label indicating presence of Pneumothorax, augmented with annotations extracted by label-free methods. The empirical evaluation shows that (1) the causal discovery method accurately recovers the underlying causal relationships and C2BM performs on par or better than baseline models, including an opaque NN, (2) C2BM achieves better performance with fewer interventions, (3) causal structure allows C2BM to mitigate the effect of spurious correlations captured by neural encoder, (4) causal structure allows C2BM to eliminate unfair bias from its decision-making.", "questions": "Questions\n1. What is the effect of data set size and causal graph complexity on the causal discovery method?\n2. While the RAG context has no impact in the Alarm domain, it improves performance on the Sachs domain. Were there any cases where using RAG degraded performance?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n1. Highly relevant problem for high stakes domains like healthcare that require interpretability and causal reliability\n2. The proposed pipeline is general enough to use more specialized LLM inference frameworks e.g., Loula et al., (2025).\n\nWeaknesses\n1. Since LLMs are stochastic and sensitive to phrasing of the prompt, the design of the prompt might affect performance.\n2. Since LLMs answer causal questions by approximately retrieving causal facts from their training data, causal graph construction would be harder in understudied domains like rare diseases.\n\nReferences\nLoula, João, et al. \"Syntactic and Semantic Control of Large Language Models via Sequential Monte Carlo.\" ICLR, 2025.", "was_revised": true, "final_justification": "The authors have addressed most of the concerns. So, I have increased my score to 5 (accept)", "rebuttal": "We thank the reviewer for their comments and feedback. As for the raised concerns and questions, individual points are addressed below.\n\nAs a side note, we found that the citation mentioned by the reviewer could support our Remark 4.1. We have decided to include it; thank you for the pointer.\n\n***W1 - Prompt design***\n\n> the design of the prompt might affect performance.\n\n**We agree**. The design of the prompt could condition the causal graph refinement. To assess this, we have added a **new ablation** study in which we fix the incomplete causal graph generated by the causal discovery algorithm, and later evaluate variants of the LLM+RAG refinement. Specifically, we evaluate the impact of four different prompting strategies with two LLMs (GPT-4o, $\\sim$200M parameters; and GPT-4o-mini, $\\sim$8M parameters). The considered strategies are as follows:\n- *minimal* prompting: simply asks the LLM to identify causal relations without additional guidance;\n- *instruction* prompting: provides a more detailed explanation of what constitutes a causal relation;\n- *few-shot* prompting: proposed in [1], combines instruction prompting with a few examples;\n- *Chain-of-Thought (CoT)*: proposed in [2], encourages the model to generate intermediate logical steps before generating the answer. This is the approach we used in the original manuscript.\n\nThe table below shows the number of Mistaken Edges for datasets with known ground-truth causal graphs (reliability of standard, flat, CBMs is also reported for reference).\n\n| LLM         | Prompting Strategy | Asia | Sachs | Insurance | Alarm | Hailfinder |\n| ----------- | ------------------ | ---- | ----- | --------- | ----- | ---------- |\n| flat CBMs   |         --         | 11   | 23    | 74        | 78    |     117    |\n| gpt-4o      | minimal            | 1    | 12    | 19        | 11    |     22     |\n| gpt-4o      | instruction        | 1    | 11    | 19        | 11    |     22     |\n| gpt-4o      | few\\_shot          | 1    | 8     | 19        | 11    |     22     |\n| gpt-4o      | CoT (paper)        | 1    | 7     | 18        | 9     |     22     |\n| gpt-4o-mini | minimal            | 1    | 10    | 19        | 10    |     22     |\n| gpt-4o-mini | instruction        | 1    | 9     | 19        | 11    |     22     |\n| gpt-4o-mini | few\\_shot          | 1    | 10    | 19        | 11    |     22     |\n| gpt-4o-mini | CoT                | 1    | 7     | 19        | 10    |     22     |\n\nDespite moderate variation in the Sachs dataset, the performance is generally very **robust** to the prompt strategy, and causal reliability is **largely superior to standard CBMs** in all cases.\n\n\n\n***W2 - Understudied domains***\n\n> causal graph construction would be harder in understudied domains like rare diseases\n\nThis is undoubtedly true, as we mentioned in our limitations (l.366). However, we believe understudied domains leave very little room for any alternative method. Note that, in any case, our pipeline can still approximate causal relations from observational data only. We hence believe C$^2$BM could be useful for exploratory research in understudied domains to restrict the search space of possible hypotheses and contribute to forming a background knowledge.\n\n\n\n***Q1 - Sample complexity of graph discovery***\n\n>  What is the effect of data set size and causal graph complexity on the causal discovery method?\n\nWe explored this with a **new sensitivity study**, running the causal graph pipeline (causal discovery + LLM refinement with CoT prompt) across all datasets with an available ground-truth graph. For each, we varied the number of data points $N$ from 100 to 10,000. \n\n| Dataset    | Flat CBMs |                |      |   C$^2$BM's   |   causal     |  graph    |                 |\n|------------|-----------|------------------------|------|------|------|------|-----------------|\n|            |           | **Data size** $\\rightarrow$          | **100**  | **500**  | **1000** | **5000** | **10000 (Paper)**   |\n| **Asia**       | 11        |                        | 6    | 3    | 1    | 1    | 1               |\n| **Sachs**      | 23        |                        | 15   | 12   | 11   | 7    | 7               |\n| **Insurance**  | 74        |                        | 47   | 35   | 24   | 22   | 18              |\n| **Alarm**      | 78        |                        | 31   | 16   | 9    | 9    | 9               |\n| **Hailfinder** | 117       |                        | 56   | 47   | 44   | 22   | 22              |\n\nA few considerations:\n\n- C$^2$BM's causal graph is consistently more causally reliable than the flat structure implicitly assumed by standard CBMs, regardless of the dataset size.\n- As expected, increasing the number of data points leads to better alignment between the estimated and true causal graphs. Causal reliability tends to remain stable at larger data sizes.\n- We observed that the data size threshold for reliable performance does not strictly depend on graph size. We speculate this is due to the varying impact of LLM-based refinement across datasets. In some cases, an effective background knowledge can compensate for limited data.\n\nWe appreciate the reviewer’s question and will incorporate these findings into the paper.\n\n\n***Q2 - Clarification on RAG performance***\n\n> Were there any cases where using RAG degraded performance?\n\nNo, we did not observe any cases where using RAG degraded performance. We speculate this is because all our investigated domains are well-supported by archived scientific literature. In the paper, we included results for all the datasets we experimented on.\n\n\nWe hope to have clarified all of the reviewer’s concerns and are happy to provide further details if needed.\n\n**Refs:**  \n[1] Ye et al., Investigating the effectiveness of task-agnostic prefix prompt for instruction following, AAAI, 2024  \n[2] Wei et al., Chain-of-thought prompting elicits reasoning in large language models, NeurIPS, 2022"}, {"review_id": "I2y3FOsVLJ", "reviewer": "Reviewer_NX3e", "summary": "The paper introduces Causally-reliable Concept Bottleneck Models (C2BMs), an extension of CBMs that force predictions to flow through an explicit causal graph model. Specifically, the authors propose a neural architecture that couples a concept encoder with a meta-model that adaptively parameterises linear structural equations. To build the causal graph, the authors propose an automated pipeline that discover concepts and edges using a retrieval-augmented LLM. They conduct experiments on multiple datasets showing comparable performance to other CBM architectures but find that C2BMs decrease reliance on spurious correlations and enable interventions to improve fairness.", "questions": "- Have you quantified the extend to which the quality of the causal graph is a determining factor for model performance?", "limitations": "See above", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strenghts:**\n- The integration of known causal relationships into CBMs is well motivated\n- The proposed architecture combining a learnable encoder with adaptive structural equations is elegant\n- The evaluations demonstrates comparable performance to other CBMs while providing better steering ability\n- The causal graph construction pipeline using LLMs can reduce reliance on human experts\n\n**Weaknesses:** \n- The causal model injects the graphs's inductive biases into the learner. Thus, if the graph is misspecified, these biases can hurt task accuracy relative to graph-agnostic models. \n- The paper only contrasts the learned graphs against a ground truth graph but does not report performance when the model is trained with random or adversarially corrupted graphs. Thus, it's unclear how sensitive C2BM is to the quality of the causal graph.", "was_revised": true, "final_justification": "The paper demonstrates how injecting domain‐level causal structure into concept bottleneck models can preserve accuracy while enabling meaningful intervention. The author rebuttal mostly addresses my main concern that a misspecified or corrupted graph might degrade performance more than in graph-agnostic baselines.", "rebuttal": "We thank the reviewer for their comments and feedback. As for the raised concerns and questions, individual points are addressed below.\n\n***W1 - Performance under adversarial graph corruptions***\n\n> if the graph is misspecified, these biases can hurt task accuracy relative to graph-agnostic models.\n\n**We disagree**. C$^2$BM is built upon a key theoretical result (see Appendix D.1), which shows that it is a **universal approximator** for the final task, **regardless** of the specific causal graph. This implies that C$^2$BM retains task accuracy of black-box models, even with a largely corrupted graph, as long as the architecture is sufficiently expressive. \nRecall that, while task accuracy is preserved, causal reliability does degrade under graph corruption. This would affect the quality of explanations and out-of-distribution (OOD) performances, as shown by the differences between standard CBMs (i.e., flat, unreliable causal structures) and C$^2$BMs in Tab. 2 and Fig. 5, respectively.\n\n\n***W2,Q1 - Sensitivity to graph corruptions***\n\n> it's unclear how sensitive C$^2$BM is to the quality of the causal graph. ... Have you quantified the extend to which the quality of the causal graph is a determining factor for model performance?\n\nDespite the theoretical argument, we agree that it is important to empirically validate C$^2$BM's robustness in task accuracy. To this end, we added a **new sensitivity analysis** evaluating task accuracy when training C$^2$BM on increasingly more corrupted graphs (3 seeds, uncertainties represent 2 sample mean $\\sigma$). We consider two strategies, both starting from the ground-truth graph. In the first, we alter a percentage $p$ of graph edges by applying one of the following operations (chosen randomly, per edge): flipping, addition, or removal.\n\n| Dataset / p    | 0.05          | 0.1           | 0.2           | 0.4           | 0.6           | 0.8           | 1.0           |\n|-------------|---------------|---------------|---------------|---------------|---------------|---------------|---------------|\n| **asia**     | 71.8 ± 0.6  | 71.6 ± 0.5  | 71.43 ± 0.6  | 71.7 ± 1.3  | 70.2 ± 1.9  | 71.4 ± 1.2  | 70.2 ± 0.9  |\n| **sachs**    | 65.1 ± 1.9  | 65.6 ± 2.0  | 64.85 ± 2.0  | 64.6 ± 1.2  | 65.5 ± 1.5  | 65.2 ± 1.5  | 65.0 ± 2.0  |\n| **insurance**| 67.2 ± 1.6  | 67.0 ± 3.1  | 67.45 ± 2.3  | 66.1 ± 1.1  | 67.0 ± 2.6  | 66.8 ± 2.6  | 67.2 ± 3.1  |\n| **alarm**    | 61.9 ± 2.6  | 61.7 ± 2.1  | 61.48 ± 2.5  | 61.9 ± 2.7  | 60.6 ± 1.9  | 60.9 ± 1.6  | 61.4 ± 1.7  |\n| **hailfinder**| 74.0 ± 1.5 | 73.1 ± 3.0  | 72.32 ± 2.2  | 72.3 ± 3.2  | 73.0 ± 2.1  | 71.8 ± 1.6  | 72.4 ± 1.4  |\n\nIn the second one, we progressively transform the graph into the flat structure assumed by standard CBMs, i.e., progressively connecting a percentage $p$ of nodes to the chosen task and eliminating other outgoing edges.\n\n\n| Dataset / p    | 0.05          | 0.1           | 0.2           | 0.4           | 0.6           | 0.8           | 1.0           |\n|----------------|---------------|---------------|---------------|---------------|---------------|---------------|---------------|\n| **asia**       | 72.0 ± 1.0  | 72.0 ± 1.0  | 71.5 ± 0.3  | 71.7 ± 0.8  | 71.7 ± 0.5  | 71.2 ± 1.6  | 71.1 ± 1.1  |\n| **sachs**      | 65.5 ± 2.2  | 64.8 ± 3.0  | 65.8 ± 1.6  | 64.7 ± 1.5  | 65.2 ± 1.2  | 65.1 ± 2.7  | 64.9 ± 2.2  |\n| **insurance**  | 67.4 ± 2.6  | 66.2 ± 1.7  | 66.3 ± 2.5  | 65.6 ± 3.3  | 65.3 ± 2.4  | 66.9 ± 1.8  | 64.7 ± 3.1  |\n| **alarm**      | 62.6 ± 2.9  | 61.6 ± 2.6  | 61.6 ± 2.6  | 60.5 ± 2.6  | 61.2 ± 1.8  | 60.6 ± 3.2  | 61.3 ± 1.9  |\n| **hailfinder** | 73.1 ± 1.1  | 73.2 ± 1.9  | 72.5 ± 0.9  | 72.7 ± 2.9  | 73.0 ± 2.6  | 73.0 ± 1.7  | 72.9 ± 1.9  |\n\nThe results confirm that task accuracy **remains stable** even when the causal graph is heavily corrupted, empirically supporting the theoretical claim. We thank the reviewer for raising this point, we will extend the paper with this new experiment. \n\nWe hope to have clarified all of the reviewer’s concerns and are happy to provide further details if needed."}, {"review_id": "t9xyAxCOQV", "reviewer": "Reviewer_mU7E", "summary": "This work attempts to bridge the gap between CBMs and Causal reliability with a new framework C2BM. It is based on performing LLM guided causal discovery on an identified concept space and combining this with structured causal modelling. The authors investigate the overall predictive performance as well as the. ability to perform concept interventions and debiasing.", "questions": "I would consider citing [1] in terms of concept interventions:\n\n[1] Steinmann, David, Wolfgang Stammer, Felix Friedrich, and Kristian Kersting. \"Learning to intervene on concept bottlenecks.\" arXiv preprint arXiv:2308.13453 (2023).", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "I generally like the idea very much of combining concept XAI with causal reliability. Also the paper is very well written and it is easy to follow. Generally, I have very few weaknesses, just a few clarification issues which it would be good if the authors could clear up my misunderstandings. I am aware that the predictive performance experimental results don't look smashing, but I don't think this should be the focus of this work, i.e. the approach is interesting enough to overlook this.\n\nMainly, I am missing where the label prediction occurs in the method description, but also within figure 1. Maybe I am not an expert enough on SCMs, but how are the classifications of the evaluations done based on the setup, i.e. how is the most likely variable selected from the causal graph bottleneck. Standard CBMs utilize a final predictor, but I am missing the information on how this is handled here.\n\nIn this context I am still a little confused about what the goal is of C2BM and how this differs from simply doing causal discovery. It seems an image is first processed into concepts and then a causal graph is discovered based on this. Could the authors elaborate on this?\n\nIn this context I am also a little puzzled by the broader impact statement \"C2BM has the potential to significantly narrow the hypothesis space in complex scientific domains where even a panel of human experts might struggle to identify or exclude plausible hypotheses worth testing. \" again isn't this \"simply\" causal discovery? I.e. why do we need C2BM for this use case?\n\nThe debiasing experiment is a little unclear to me. How are the interventions being done here? I would assume if a vanilla CBM has color and shape concepts it should be very easy to debias the model, by reducing the reliance on the spurious concept. Could the authors clarify how exactly the results of figure 5 were created, particularly for the baseline models?\n\nI'm a little surprised the effect of C2BMs in Fig. 4 is quite low (often only a few decimal point advantages). I would expect particularly here such models to shine. Could the authors discuss a little bit on this. Do these datasets simply don't require such interventions?\n\nTab. 1 is very squeezed. Do you think it might be an idea to move two columns to the appendix to improve the readability within the main text?", "was_revised": true, "final_justification": "My clarification questions have been addressed. I have skimmed over the other reviews, but haven't seen any major issue which would change my view of my score.", "rebuttal": "We thank the reviewer for their comments and feedback. We particularly appreciate the positive remarks on the novelty, soundness, and clarity of our work. As for the raised questions, individual points are addressed below.\n\n***Q1 - Label prediction***\n\n> I am missing where the label prediction occurs in the method description\n\nStarting from a data sample (e.g., raw image), we first extract a latent representation using a neural encoder. Then, an MLP is used to predict the probability of each root concept being in a specific state (e.g., true/false). Then, the state probabilities for all descendant concepts (including the task concept) are obtained by **iteratively applying Eq.2**. Specifically, Eq.2 shows that the probability of each concept state is determined by a linear combination of the state probabilities of its parents in the causal graph. Notice that, once the causal graph is determined and C$^2$BM is trained, one can choose any endogenous variable (concept) as the query (task).\nWe refer to Fig. 7 in the Appendix for a more detailed visualization.\n\nWe recognize the paper speaks in terms of 'variables' $V_i$ rather than state probabilities. This could be ambiguous. To improve clarity on this aspect, we added a footnote in l.193: \"We refer to $V_i$ as a variable to allow for a more general formalization. In a classification setting, concepts assume categorical values; hence $V_i$ represents the probability of a concept state activation.\"\n \n\n\n\n***Q2 - Difference with causal discovery***\n\n> the goal is of C$^2$BM and how this differs from simply doing causal discovery\n\nIt is important to distinguish C$^2$BM from the components that need to be determined **prior** to its training and functioning. C$^2$BM is not a causal discovery algorithm. It is an inference ML model (see right side of Fig.1) trained on a given causal graph of concepts. For each data sample, the model learns to predict the activation of each variable in the graph based on its causal parents. In this setup, the causal graph acts as a structural prior, effectively a hyperparameter, which can be obtained from expert knowledge and/or from (potentially separate) data using causal discovery methods (on this, see Remark 4.1 and response Q2 to Reviewer wNgz).\n\nTo give a concrete example: causal discovery might be used by a hospital to infer which factors are, in general, direct causes of a disease (as in our *Pneumothorax* dataset). C$^2$BM, once trained on this structure, can then make patient-specific predictions, e.g., estimating whether a particular patient is sick and which of the known causes are likely active in their case.\n\n\n***Q3 - Clarification on broader impact***\n\n> I am also a little puzzled by the broader impact statement \"C$^2$BM has the potential to significantly narrow the hypothesis space in complex scientific domains where even a panel of human experts might struggle to identify or exclude plausible hypotheses worth testing.\"\n\nWe thank the reviewer for pointing this out; we recognize this sentence could be unclear. The sentence highlights a benefit commonly associated with causal discovery, one that C$^2$BM both **inherits and extends**. While causal discovery narrows the hypothesis space by uncovering candidate causal structures, C$^2$BM builds on top of this and enables predictive inference over those structures. That is, C$^2$BM allows for interpretable, concept-level predictions grounded in those relationships. In this way, C$^2$BM preserves the scientific value of causal discovery, while adding the practical ability to make predictions from data and unstructured knowledge, and generate explanations, critical in real-world decision-making contexts.\n\n\n***Q4 - Clarification on debiasing experiment*** \n\n> I would assume if a vanilla CBM has color and shape concepts it should be very easy to debias the model\n\nWhile, in principle, a standard CBM could support debiasing by manually setting the appropriate weights to zero, this requires knowing which relations are spurious in advance. In practical applications, this could require substantial domain expertise. In contrast, our pipeline incorporates a causal graph construction component that can **automatically detect and remove spurious correlations** from unstructured knowledge. C$^2$BM's inference flows through such graph that is already debiased, and hence does not require manual post-hoc debiasing.\n\n> How are the interventions being done here? ... Could the authors clarify how exactly the results of figure 5 were created\n\nThe interventions in this experiment follow the standard practice in concept-based explainability [1]: we replace the predicted concept value with its ground-truth state and measure the consequent improvement in the task accuracy.    \n\nSpecifically, all models were trained on a biased training set (color $-$ parity) and evaluated on a test set where the spurious correlation is reversed. As expected, all models perform poorly ($\\sim 0$%) when tested on reversed data. However, unlike the baselines, C$^2$BM is able to recover task accuracy when we intervene on the concept that is causally related to the task (number $\\rightarrow$ parity). The other models fall short in doing so, since they still rely on spurious correlations.\n\n\n***Q5 - Interventional accuracy***\n\n> I would expect particularly here such models to shine. Could the authors discuss a little bit on this.\n\nWe speculate these are primarily due to the probabilistic nature of our datasets. In such settings, fixing the state of the parents is often **not sufficient** to uniquely determine the state of a child. **Exogenous factors also matter**. Therefore, achieving large improvements in intervention performance in these settings is inherently difficult. As shown in Fig.4, even baseline models exhibit small gaps between them. That said, our model consistently improves over all baselines across all datasets, highlighting an advantage even under these constrained conditions.\n\nAs proof of this, in concept-complete settings such as the synthetic $CUB_c$ dataset, where we explicitly made a few child nodes deterministically dependent on their parents, our approach yields significantly larger gains.\n\nWe see our work as a first step toward integrating causal reasoning with concept-based interpretability. We believe incomplete or noisy causal structures can offer a promising direction for future research.\n\n\n\n***Q6 - Table formatting***\n\n> Do you think it might be an idea to move two columns to the appendix to improve the readability within the main text?\n\nWe included incomplete datasets because we were concerned that using only the complete datasets would not clearly convey the message we aimed to communicate regarding preserving the black-box task accuracy.\nWe are considering removing cMNIST, as it is a very simple task, in order to improve readability.\n\n\n***Q7 - Citation recommendation***\n\n> I would consider citing [1] in terms of concept interventions\n\nWe thank the reviewer for the suggestion, we think it is relevant to highlight the relevance and timeliness of interventions. We have added the reference at l.220.\n\n\nWe hope to have clarified all of the reviewer’s concerns and are happy to provide further details if needed.\n\n\n**Refs:**  \n[1] Koh et al., Concept bottleneck models, ICML, 2020"}]

Author rebuttal:

Meta-review: This paper extends concept bottleneck models (CBMs) by introducing a causal structure over concepts, producing causally reliable CBMs (C2BMs). A pipeline is proposed to automatically learn such structures from observational data and background knowledge via LLMs. Experiments suggest that C2BMs improve interpretability, causal reliability, and intervention responsiveness while maintaining accuracy.

**Strengths**

* Tackles an important and timely problem at the intersection of interpretability and causality.

* Elegant architecture combining concept encoders with structural causal equations.

* Pipeline to integrate observational data and LLM-based causal discovery is novel and flexible.

* Demonstrates causal reliability, intervention capacity, and fairness benefits compared to standard CBMs.

Authors provided extensive rebuttal clarifications, new ablations, sensitivity studies, and updated results (including with GPT-5).

**Weaknesses**

* Initial clarity issues: some reviewers found the description of bottleneck evaluation, training details, and role of causal discovery confusing.

* Concerns about empirical scope and sensitivity to graph misspecification, though rebuttal added robustness experiments addressing this.

* Improvements in accuracy over baselines are modest; value lies primarily in reliability and interpretability.

The consensus among engaged reviewers leans toward acceptance, with clear appreciation of the contributions despite some limitations.

**Recommendation**
This work makes a solid contribution by bridging CBMs with causal modeling and proposing an automated, LLM-based pipeline for structure discovery. While empirical gains in accuracy are limited, the advances in interpretability, reliability, and intervention justify acceptance.

Final recommendation: Accept.
Decision: Accept (poster)