Bayesian Network Structure Discovery Using Large Language Models
Authors: Yinghuan Zhang
Yufei Zhang
Parisa Kordjamshidi
Zijun Cui
Venue: Accepted by TMLR ()

Abstract: Understanding probabilistic dependencies among variables is central to analyzing complex systems. Traditional structure learning methods often require extensive observational data or are limited by manual, error-prone incorporation of expert knowledge. Recent studies have explored using large language models (LLMs) for structure learning, but most treat LLMs as auxiliary tools for pre-processing or post-processing, leaving the core learning process data-driven. In this work, we introduce a unified framework for Bayesian network structure discovery that places LLMs at the center, supporting both data-free and data-aware settings. In the data-free regime, we introduce \textbf{PromptBN}, which leverages LLM reasoning over variable metadata to generate a complete directed acyclic graph (DAG) in a single call. PromptBN effectively enforces global consistency and acyclicity through dual validation, achieving constant $\mathcal{O}(1)$ query complexity. When observational data are available, we introduce \textbf{ReActBN} to further refine the initial graph. ReActBN combines statistical evidence with LLM by integrating a novel ReAct-style reasoning with configurable structure scores (e.g., Bayesian Information Criterion). Experiments demonstrate that our method outperforms prior data-only, LLM-only, and hybrid baselines, particularly in low- or no-data regimes and on out-of-distribution datasets.

Published in Transactions on Machine Learning Research (01/2026)
Bayesian Network Structure Discovery guage Models
Using
Large
Lan-
Yinghuan Zhang Independent Researcher
yingkuan.fashh@@maallco
Yufei Zhang Independent Researcher
yuferihhang06kknttlaok.omm
Parisa Kordjamshidi
kordjamsSmmsu.cde
Department of Computer Science and Engineering Michigan State University
Zijun Cui
cuisijun Settsjju@@mm...o Gmsu. edu
Department of Computer Science and Engineering Michigan State University
Reviewed on OpenReview: https: // openreview. net/ forum?? id= G4mrOBLVV Via
Abstract
Undorstanding probabilistic dependencies among variables central to analyzing complex systems. Traditional structure learning methods often require extensive observational data are limited by manual, error-prone incorporation of expert knowledge. Recent studies have explored using large language models (LLMs) for structure learning, but most treat LLMs as auxiliary tools for pre-processing or post-processing, leaving the core learning process data-driven. Tn this work, we introduce a unified framework for Bayesian network structure discovery that places LLMs at the center, supporting both data-free and data- aware settings. In the data-free regime, we introduce PromptBBN, which leverages LLM reasoning over variable metadata generate complete directed acyelic graph (DAG) in single call. PromptBN effectively enforces global consistency and acyclicity through dual validation, achieving constant O(1) query complexity. When observational data are available, we introduce ReActBN to further refine the initial graph. ReActBN combines statistical evidence with LLM by integrating novel ReAct-style reasoning with configurable structure scores (c.g., Bayesian Information Critcrion). Experiments demonstrate that our method outperforms prior data-only, LLM-only, and hybrid baselines, particularly in low- or no-data regimes and on out-of-distribution datasets.
Code is available at http://hhhbb.ccm//errrrg//////
1 Introduction
Structure discovery of probabilistic graphical models (PGMs) key to understanding complex systems, it reveals the underlying dependency structure among variables. Traditional approaches, including constraint- based algorithms (Spirtes et al. 2001) and score-basod methods (Chickering 2002b), typically require large amounts of observational data and intensive computation, limiting their practical use. Incorporating domain knowledge is another option, but it is typically done manually, which is labor-intensive and error-prone. This motivates the development of new methods to infer probabilistic structures more effectively and efficiently.
Recently, large language models (LLMs) trained on largo-scale text corpora have demonstrated remarkable performance (Brown) 2020), including capabilities in probabilistic reasoning and the exctinn of structured

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knowledge from text [Nafar et al. 2024 2025a b). Their ability to capture a broad range of dependencies,
including implicit causal and statistical structures, makes LLMs promising tool for structure learning. Sev- eral approaches attempt to use LLMs as sources of prior knowledge (Ban et al. 2025 Vashishtha et al. 2023 Long et al. 2023a) or as post-hoc verifiers (Khatibi et al. 2024). In both cases, the core structure learning process constrained to be lightly coupled with data-driven optimization techniques. Directly eliciting full probabilistic graphs from LLMs remains challenging, mainly due to two factors: (1) the combinatorial complexity of graph construction, making it difficult for LLMs to generate globally coherent structures, and (2) the lack of effective mechanisms to enforce structural constraints (e.g., acyclicity) and to incorporate observational data within prompt-based interactions.
In this paper, we address these challenges by introducing a unified two-phase LLM-centered framework for Bayesian network (BN) structure learning. The first phase, PromptBN, directly elicits a coherent and acyclic directed acyclic graph (DAG) through promppro strategy with dual validation that operates entirely without observational data. When data are available, the second phase, ReActBN, performs iterative refinement. in Reason-and-AAd (ReAct) workflow (Yao et. al. 2023), where oflline-computed structure scores (e.g., BIC) evaluate candidate neighbors and guide the model's decisions. Unlike prior LLM-based methods that delegate structural correction to external algorithms, our method maintains the LLM in the decision loop across both phases, enabling a more integrated and interpretable approach to BN discovery. Our key contributions are as follows:
We propose an LLM-based framework for flexible structure discovery with or without observational data.
We propose a meta meta-prompting approach that elicits a Directed Acyclic Graph (DAG) in one LLM inference with (1)) query complexity.
We propose novel agentic search framework to refine the initial structure by integrating Reason- and-Act (ReAct) for LLM internal knowledge reasoning with structure scores like BIC.
We perform extensive experiments on diverse datasets, showing superior structure discovery perfor- mance over state-of-the-art methods. In specifie, contamination tests show that the observed gains arise from reasoning rather than memorization. Complexity analyses further demonstrate scalability to large, complex graphs, especially compared with existing LLM-based approaches.
2 Related Work
Data-driven structure learning Conventional structure learning methods infer Directed Acyelic Graphs (DAGs) primarily from observational data, broadly categorized into construint-based and score-based ap- proaches (Glymour et al.] 2019 Scutari et al. 2019). Constraint- based methods, such as the Peter-Clate algorithm (Spirtes et al. 2001 and its variants, identify probabilistic structures by assessing conditional independencies. Score-based methods, including Hill Climbing (Tsamardinos et al. 2006) and Greedy Equiv- alence Search (Chickering. 2002ah, optimize predefined scoring functions (e.g., Bayesian Information Crite- rion). Although these methods are effective with sufficient data, their performance deteriorates significantly with limited samples (Cui et al. 2022a). Rescarchers incorporate prior expert knowledge, such as structural constraints or topological ordering, inlo learning frameworks (Borboudakis & Tsamardinos 2012 Constanti- nou et al. 2023 Chen et al.) 2016) when data is insufficient. Eliciting expert knowledge, however, remains challenging and labor-intensive.
LLM-guided structure discovery with data Recent methods explore LLMs in structure discovery, primarily as complement to conventional approaches. Several researchers use pairwise prompting to guide
LLMs in uncovering causal relationships between pairs of nodes (Choi et al. 2022 Kıemman et al. 2023 Long ct al., 2023b). On top of that, Jiralerspong et al. (2024) propose a Breadth- approach, in which the LLM prompted to find all connected nodes given the current node. Pearson correlation coefficients also included in the prompts hints.
Search (BFS) prompting
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In addition, LLMs have been used to provide structural priors, establish topological orderings, refine learned probabilistic graphs post hoc (Vashishtha et al. 2023 Long et al. 2023a Khatibi et al. 2024). Some methods also use LLM-generated statements to iteratively guide data-driven approaches (Ban et al. 2023;
Chen et al. 2023) or to incorporate observational data into prompts under both pairwise and BFS paradigms for hurther improvement (Susanti & Färber 2025). Beyond pairwise- and BFS-based prompting, Ban et al. (2025) introduced LLM-CD, hybrid approach that treats the LLM as a structured knowledge retriever elicit graph-level priors and structural constraints. Once these LLM-derived priors are established, a classical data-driven structure learning algorithm can be applied to conduct the subsequent search.
LLM-only structure discovery without data LLM-only structure learning has been explored two main directions. Cohrs et al. (2024) propose an LLM-driven conditional independence (CT) testing procodure embedded within the PC framework. Although conceptually appealing, the repeated CI queries make the method computationally expensive and limit its applicability small networks. In parallel, Babakov et al.) (2024) investigates multi-expert strategy in which facilitator LLM generates multiple personas and ag- gregates their proposed causal relations. Although this design excels with ((1) query complexity effectively, it still admits the worst-case O(N2) query complexily when resolving conflicting directed pairs.
These limitations leave open the need for a more efficient and integrative approach. Our method addresses this gap by directly cliciting full causal graph through a single structured prompt-achieving O(1) query complexity and by offering unified framework that naturally exlends to dala-aware refinement. when observational data are available.
3 Problem Definition
A Bayesian network (BN) G is a directed acyclic graph (DAG) consisting of nodes V and edges E, i.e., G = (),E). The task of structure learning for BNs typically refers to learning the structure of a DAG from the data D. For example, score-based structure learning can be formulated as a constrained optimization problem:
G arg max Score(G, geG
(1)
s.t. G E DAG
where 9" is the optimal network structure; G is the search space of all possible DAGs; Score(G, D) is scoring function that evaluates how well the structure & fits the data D.
4 Proposed Approach
We have developed two-phase framework for LLM-centered discovery of BN structures.
Phase-1 PromptBN: A single-step prompting approach that elicits Bayesian network structures directly from variable metadata, relying exclusively the LLM's internal knowledge basc
Phase-2 ReActBN: A ReAct-inspired agent system that integrates score-based evaluation with LLM reasoning, where the LLM supervises and selects optimal actions from top-k candidates identified through scoring mechanisms.
The framework overview can be found at Figure 1 and pseudocode is available at Appendix
4.1 PromptBN: Bayesian Network Estimation via Prompting
In Phase 1, we formulate structure learning as a single-query inference task:
..I.M-Querr (eeuurrmmeaaaaa(()) Mn)oledge(()))
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Phase-1 PromptBN
Reasoning Prompt
Bayesian Network Output
Variables Metadata
Bayasian
Pollution
Whether the patiant was exposed to polluted air or not Whether the patient is
T/F
"bn": { "nodes"; II "node_id"; cintegers, "noee_nane": estring,, " "parents": [estrings, ...l,
Pollution
Smoker
Smoker
T/F
Whether the patient ls suffering from lung cencer or not Denotes the from chest
Cancer
Cancer
Response Structure
"edges": I "foo""; estring,, "to": estringp, "justification's estrings
Xray
Xray
Dysp
Whether the patient is suftering T/F from Dyspnoea
Dysp
Output
two graph reprosentation match. generated graph have
comNec wrng
Orick
Phase-2 2 ReActBN
Constraint
Phase-l Output as Start Graph
ReAct-based Search
Refined Bayesian Network
Action Space
Observation Top-k Observations from
Action Space
delete
"Smoker" "Dysp"
The goal
Pollution
Smoker
Removing the edge Smoker -> Dysp suggests that, according the data, the direct dependency from Smoker Dyspnoea is not supported and its removal improves the model's fit the data without unnecessary complexity.
delete "Pollution" -> "Xray" à 0.00
Cancer
Reasoning
flip
"Smoker"
"Canoer"
4
Xray
Dysp
add
"Pollution" >> "Smoker" -1.90
termination
Action
Delete Smoker +> Dysp
Figure 1: Overview of the proposed unified prompting framework for Bayesian network structure discovery using LLMs. The framework supports both data-free and data-driven settings through a two-phase process: Phase 1 induces the structure without observational data, while Phase refines the estimated structure by incorporating data when available.
where L.M--Qerr(()) denotes single query to the selected model M, leveraging its implicit internal knowl- edge. Here, V represents the set of variables, and the metadata of each variable include its name, description of the variable, and distribution of the variable. An example input is illustrated in Figure [
Our approach employs meta-prompting (Zhang et al. 2023), where the prompt defines not only the task objective but also the reasoning protocol, response structure, and output constraints. Specifically, PromptBN requires the model to produce a valid Directed Acyclic Graph (DAG) while articulating its reasoning process.
To ensure robustness, require the model to output the graph structure in two complementary represen- tations: node-centric format that specifics parent nodes for each variable, and an edge-centric format that enumerates all directed edges as "from-to" pairs.
Following generation, we apply dual validation procedures. First, Structural Consistency Validation confirms that parent-child relationships are identical across both representations. Second, DAG Constraint Validation verifies acyclicily. If the output fails either criterion, prompt the LLM to regenerate the structure until validation succeeds or the maximum retry threshold is reached.
4.2 ReActBN: ReAct-based Score-Aware Search
As ratustrated in Fig. Phase 2 refines the initial Bayesian network (BN) structure obtained in Phase by incorporating observational data. This phase adopts ReAct-style (Reason + Act)) iterative framework YYo ct al. 2023 referred to as ReActBN, which an LLM explicitly reasons about candidate structural modifications and selects actions based on statistical feedback.
Iterative Search Framework. Starting from the current directed acyclic graph (DAG) G., ReActBN performs iterative search procedure. At each reasoning iteration, ReActBN presents the LLM with the
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top-k candidate structures. These candidates are derived through exhaustive enumeration of all valid neighboring structures, each annotated with its structure score and score differential.
Structure Scoring. The scoring function itself is fully configurable to accommodate different statistical criteria. Each candidate structure G' is evaluated using a score-based criterion. In this work, we employ the Bayesian Information Criterion (BIC) defined as
dim(G) 2
BIC(G D) = log G. (c)c))
log N,
(2)
where denotes the observational dataset, O are the maximum likelihood parameter estimates associated with G, (G) the number of free parameters in the network, and N is the sample size. For each candidate G", we compute the score difference Agr BIC(G") BIC(G.).
LLM-Guided Reasoning and Action Selection. At cach iteration, the LLM is provided with com- prehensive contextual information, including variable meladala, the current graph structure G,, its BIC score, the set of top-k candidate actions with their associated score differences, and tabu list of previously visited structures. Unlike traditional groedy score-based methods (c.g.. Hill Climbing or Greedy Equivalence Search), which deterministically select the single highest-scoring move, ReActBN allows the LLM to reason over multiple candidate actions. The L.M outputs a selected action (or a termination decision), accompa- nied by an explicit reasoning trace and confidence estimate. To encourage exploration and prevent cycling, we incorporate tabu search mechanism that prohibits revisiting previously explored structures.
5 Experiments
In this section, we validate the effectiveness of our approach. We first describe the datasets, evaluation metrics, and experimental setup (Section 5.1 We then compare our method against existing approaches in both data-free and data-aware scttings classical (Section 5.2) and newer (Section 5.3 benchmarks, followed by valuable study of potential contamination in LLMs (Section 5.4h.
5.1 Evaluation Datasets and Metrics
Datasets We evaluate ten datasets for Bayesian network structure learning. Specifically, seven datasets, including Asia, Cancer, Earthquake, Child, Insurance, Alarm, and Hailfinder, are widely used benchmarks in structure learning, from Bulearn (Scutari 2010). The remaining three datasets, blockchain, consequence- Covid (referred to covid hereafter), and disputed3, are recent datasets from BnRep [Leoneli 2025). We retrieve all the metadata, including variable names, descriptions, and distributional properties, from the released packages, except for Insurance, whose description taken from Long et al. (2023a). For the Earth- quake dataset, since no quantitative results are available for comparison, we report qualitative evaluation results by visualizing the learned structure. We provide a summary of all dataset statistics in Appendix B.1.
Evaluation metrics We evaluate both the effectiveness and efficiency of structure learning algorithms. For effectiveness, use two well-established metrics: Structural Hamming Distance (SHD) and Nor- malized Hamming Distance (NHD). To compute SHD, both the predicted graph and the ground-truth DAG are first converted into their corresponding completed partially directed acyclic graphs (CPDAGs). SHD is then defined as the number of edge modifications, including insertions, deletions, or reversion, re- quired to make the predicted CPDAG identical to the ground-truth CPDAG. NHD a variant of SHD that offers scale-invariant measure of structural difference. Following Khatibi et al. (2024). for a graph with 1 N nodes, NHD is calculated as > Ey= N² 1(E, * Eny), where Fay and Egy denote the presence or absence of an edge from node i to node j in the ground truth and predicted graphs, respectively. 1(.) is the indicator function, which returns 1 if E., # E.j and otherwise.
Experimental setup. 1) LLM models: Unless otherwise specified, we use the OpenAI model 03-2025-04-16 (referred to o3 hereafter) for PromptBN and gpt-4. 1-2025-04-14 for ReActBN, accessed
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via the OpenAI API (Achiam et al. 2023). We select o3 for PromptBN due to its strength in structured multi-step reasoning for single-shot DAG generation, and gpt-4.1 for ReActBN due to its reliable stepwise instruction following for iterative refinement. 2) Baselines: We compare approaches across three categories: LLM, Data, and Hybrid. The LLM category contains methods that rely on large language models without
using any observational data, including PairwisePrompl (Kiemnan et al. 2023 BFSPromppl (Jiralerspong
ct al. 2024). Scalability (Babakov et al. 2024),, and ChatPC (Cohrs et al. 2024]. The Data category consists of classical struetural-earninn algorithms that operate purely on observational samples through statistical estimation, including Hill Climbing (HC), Peter-Clark algorithm (PC-stable), Greedy Equivalence Search (GES), and three algorithms designed for low data observations (Cui et al. 2022b), i.c., RAI-BF, rT=B, and rB The Hybrid category encompasses approaches that combine LLM reasoning with observational data, including LLM-CD (HC) (Ban el al., 2025) and
(Susanti & Pär- ber. 2025 When reproducing results, we use the default parameters in the Python pgmpy package (Ankan & Textor 2024) for HC, PC-stable, and GES. We use the same sample size for all LLM-basod methods except ChatPC and Scalability, for which reproduction is not feasible. 3) Sample size: For the data-aware setting, we use 100 samples following Cui et al.] (2022a), unless otherwise specified. 4) Hyperparametens: For ReActBN, the hyperparameters are listed in Table g 5) Repeated experiments and valid runs: To address the intrinsic stochasticity of large language models, we report results over five valid runs, where a valid run is defined as one that successfully completes and produces a graph passing our dual-validation checks. We continue running experiments for each configuration until either: (1) five valid runs have been collected, or (2) five consecutive invalid runs occur, at which point we terminate that configuration.
5.2 Evaluation on Classical Benchmarks
We evaluate the performance of our proposed methods, PromptBN and ReActBN, and compare them with existing methods across the three calegories of baselines: LLM, Data, and Hybrid approaches. We present the SHD and NHD evaluations on six widely used classical Bayesian network structure learning benchmarks (Asia, Cancer, Child, Insurance, Alarm, Hailfinder) in Table
Comparison to LLM-only methods (LLM) Among LLM-only methods for Bayesian network structure discovery (PairwisePrompt, BFSPrompt, and Scalability with GPT-4), our proposed PromptBN demon- strates the most robust and consistent performance. PromptBN achieves exact recovery on Asia (SHD- = 0), high fidelity on Cancer (SHD = 0.6), and the top result on Insurance (SHD = 35.6). Even on larger networks such as Alarm and Hailfinder, PromptBN consistently attains either the best or second-best SHD and NHD. Furthermore, the performance of PromptBN+HC indicates that PromptBN provides substan- tially strouger initialization than an empty graph, improving downstream hill-climiing optimization across multiple datasets. For instance, on Hailfinder, PrompiBN+HCC reduces the SHD from 72 (HC alone) to 63.8.
As CCatPC (Cohrs et al. 2024) only reports qualitative evaluat ions without quantitative metrics, we visualize the structures produced by their method and ours in Figure 2 for comparison. On benchmarks such as Asia and Cancer, PromptBN yields noticeably cleaner and more faithful graph structures that exactly match the ground truth, underscoring the reliability of its global structure- geneerationn mechanism.
Comparison to Data-only Baselines (Data) and Hybrid Methods (Hybrid) As illustrated in Table 0 classical data-only algorithms, including PC-Stable, HC, and GFS, typically achieve near-optimal accuracy on small networks but show rapid performance degradation on medium- and large-scale benchmarks such as Insurance and Hailfinder. Low data-oriented variants, such as RAI-BF, rI and r rBFa (Cui et al., 2022b), may provide improvements selected datasets but suffer from inconsistent performance overall. By contrast, hybrid methods achieves the enhanced performance when combining LLM outputs with data. Across all data-available settings, our proposed ReActBN achieves the strongest performance on nearly all classical benchmarks. ReActBN exactly recovers the Cancer network (SHD = 0) and attains the best results on out of datascts. On Insurance, ReActBN outperforms all basclines with an SHD of 40.2. On Hailfinder, it surpasses multiple specialized low-data algorithms, including r and rBFx, demonstrating its robustness in challenging high-dimensional scenarios.
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Table 1: Evaluation on classical (Asia, Cancer, Child, Insurance, Alarm, Hailfinder) and newer (blockchain, covid, disputed3) Bayesian network datasets
Asia
Cancer
Child
Insurance
Alarm
Category #Smmple Algorithm
SHD, NHD
SHD4 NHD
SHD
NHD|
SHD,
NHDJ
SHD/ NHD
LLM
Prompss
5.6 0.072
0.0 0.000
19.6
0.042
37.6
0.044
60.0
0.044
L.M
HFSPrompt
0.0 0.000
0.0 0.000
20.4
0.047
48.4
0.058
36.2
0.026
LLM
Scalability (GPT-4)O
52.0
60.0
LLM
PromptBN (Ours)
0.0 0.000
0.6 0.024
21.6
0.045
35.6
0.044
41.8
0.031
Data
100
PC-Stable
9.0 0.156
4.0 0.160
20.0
0.065
51.0
0.073
46.0
0.037
Data
100
HC
8.0 0.125
4.0 0.160
20.0
0.065
54.0
0.067
50.0
0.040
Data
100
GES
8.0 0.125
4.0 0.160
40.0
0.110
72.0
0.099
50.0
0.042
Data
100
RAI-BFII
5.7
4.1
30.4
54.9
48.4
Data
100
r/ X
13.3
4.7
21.6
48.9
44.5
Data
100
rBFas [2]
7.7
4.1
24.2
50.1
42.7
Hybrid
LLM-CD(HC)
0.0 0.000
0.0 0.000
40.0
0.093
65.0
0.089
22.0 0.219
Hybrid
100
PairwisePrompt Data Data
0.8 0.013
0.0 0.000
20.6
0.048
0.055 444
69.4 0.051
Hybrid
100
BFSPrompt++Dat
0.4 0.006
0.2 0.008
24.4
0.054
50.4
0.061
42.0
0.030
Hybrid
100
HC
8.0 0.094
0.0 0.000
18.2 0.030
46.6
0.055
43.6
0.028
Hybrid
100
CompptN++Randonne
5.6 0.084
1.2 0.056
23.4
0.053
43.4
0.056
43.0
0.031
Hybrid
100
ReActBN (Ours)
6.4 0.075
0.0 0.000
18.0 0.028
40.2
0.049
35.4
0.024
blockchain SHD, NHD.
Hailfinder
covid
disputed3
Category #Sample Algorithm
SHD NHD) SHD NHD] SHD, NHD]
LLM
PairwisePrompt
551.0 0.176
37.4
0.260
82.6
0.433
41.2
0.057
LLM
BFSPrompt
243.0 0.077
29.0
0.201
74.6
0.401
21.6
0.027
LLM
Scalability (GPT-4)D
68.0
LLM
PromptBN (Ours)
76.8 0.025
15.2
0.100
45.0
0.225
15.6
0.018
Data
100
PC-Stable
76.0 0.024
13.0
0.104
36.0
0.160
35.0
0.069
Data
100
HC
72.0 0.024
13.2
0.094
32.6
0.156
30.8
0.051
Data
100
GES
155.0 0.050
13.0
0.083
29.0
0.142
33.0
0.053
Data
100
RAI-BEI
15.7
62.2
36.3
Data
100
rI X
88.07
14.6
47.2
34.7
Data
100
rBFas [2]
98.3f
13.6
33.6
33.4
Hybrid
LLM-CD(HC)
150.0 0.045
12.0
0.090
34.0 0.169
16.0
0.026
Hybrid
100
Pertwsssrromp--DDe Data
356.0
0.114
27.2
0.188
69.4
0.362
35.6
0.047
Hybrid
100
BoSPrompt-+D Data
140.0
0.441
18.6
0.124
65.6 0.337
24.6
0.032
Hybrid
100
PromptBN+ HHC
63.8 0.018
12.2
0.085
33.2
0.156
15.4
0.019
Hybrid
100
PromptNN+Rane Random
79.2 0.026
Hybrid
100
ReActBN (Ours)
75.0 0.023
11.0 0.074
32.8 0.164
12.2 0.015
[1] Reported in Bahakov et al. 2024p.
[2) Reported in Cui et al. 2022b)
1: the sample size is 500 instead of 100 for Hailfinder.
5.3 Evaluation on Newer Datasets
We evaluate on three recently introduced Bayesian networks (Lconelli 2025), including blockchain, covid, and disputed3 to assess the generalization capability of our method. These datasets differ substantially from the widely used classical benchmarks and are far less likely to appear in model pre-training corpora. They therefore provide a meaningful proxy for out-of-distribution (OOD) evaluation. As no prior work reports results on these networks, we reproduce all baselines and summarize the results in Table I For fair comparison, the data-driven and hybrid baselines use 100 observational samples.
Across all three networks, both PromptBN and ReActBN demonstrate strong generalization performance. Under the frte-free setting, PromptBN consistently achieves the best results, exhibiting a substantial per- formance improvements over existing LLM-only methods. For example, on covid, PromptBN attains
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asia
tub
xray
Earthquake
Burglary
Smoker
Pollution
either
lung
Alarm
Cancer
smoke
dysp
Dyspnoea
Xray
MaryCalls
JohnCalls
bronc
(i) CCatPC
(i) chatPC
(l) chatPC
asia
tub
xray
Earthquake
Burglary
Smoker
Pollution
either
lung
Alarm
Cancer
smoke
dysp
Dyspnoea
Xray
MaryCalls
JohnCalls
bronc
(ii) PromptBN / GT
(ii) PromptBN / GT
(il) PromptBN GT
(a) Asia
(b) Earthquake
(c) Cancer
Figure 2: Qualitative comparison between our method (PrompiBN) and chatPC across three benchmark datasets. Each group shows chatPC (top) and FromptBN/Groone /Ground Truth DAG (bottom): (a) Asia, (b) Earthquake, and (c) Cancer.
SHD of 45.0, oulperforming PairwisePrompl (82.6) and BFSPrompt. (74.6) by a wide margin. This high- lights PromptBN's ability to leverage metadata effectively without relying on any observational data, even in OOD scenarios. Morcover, under the data-aware sctting, ReActBN achieves the strongest overall perfor- mance, obtaining the best results on of the three networks (blockchain and disputed3), surpassing all LLM-only, data-only, and hybrid baselines. Even covid, ReActBN remains competitive over LLM-bbssd methods, showing its robustness under distribution shift and its advantage in balancing statistical evidence with LLM-driven structural reasoning. These findings demonstrate that PromptBN excels in zero-shot gen- eralization, while ReAct provides state-of-the-art performance when even modest data available. Both methods serve as strong and complementary solutions for structure learning in out-of-distribution regimes.
5.4 Contamination Tests and Analysis
Table 2: Contamination test with different forms of metadata input to LLMs
Asia
Cancer
Child
Insurance
Alarm
Hailfinder
Category #Sample Algorithm
SHD, NHD. SHD, NHD]
SHDI NHD) SHD, NHD] SHDD NHD4
SID,NDD
LLM
VarName
0.0 0.000
0.7 0.030
27.3
0.050
41.3 0.050
43.8 0.030
118.7 0.040
LLM
VarDescription
0.0 0.000
0.3 0.010
31.3
0.070
44.3
0.060
43.8
0.030
119.8 0.040
LLM
0
ScrambleOrder
0.0 0.000
0.6 0.024
24.8
0.051
37.4
0.044
48.4 0.964
97.8 0.032
LLM
PromptBN (Ours)
0.0 0.000
0.6 0.024
21.6 0.045
35.6
0.044
41.8
0.031
76.8 0.025
VarName: only variable names provided VarDescription: only variable descriptions are provided
ScrambleOrder: full metadata given but in randomly permuted order
Since the training corpora of modern LLMs are undisclosod, potential contamination cannot be fully ruled out. To better understand whether an LLM might recover underlying structures through memorization rather than genuine reasoning, conduct. three controlled contamination tests in which the LLM receives

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deliberately restricted or perturbed form of metadata and is asked to generate Bayesian network using the same PromptBN instruction template. We present the experimental results in Table 2
The contamination tests demonstrate that PromptBN's effectiveness is not attributable to memorization of dataset-specific patterns. When variable names (VarName) or descriptions (VarDescription) are used in isolation, structural recovery degrades sharply across all non-trivial networks-foo example, Child (SHD 27.3 31.3), Insurance (41.3 44.3), Alarm (43.8), and Hailfinder (118.7 119.8). Such high errors indicate that the LLM cannot reconstruct the underlying causal structures from limited or incomplete metadata, These results provide strong evidence that PromptBN is not retrieving memorized graph templates, but is instead relying on genuine relational reasoning using the full set of metadata. Morcover, after permuting all variables and removing any positional or formatting cues (ScrambleOrder)) the LLM can recover structures with SHD values that remain close to those of PromptBN for instance, Child (24.8 vs. 21.6) and Insurance (37.4 vs. 35.6). Such stability under order perturbation confirms that PromptBN does not depend on memorizing any specific ordering of metadata for its predictions. Instead, it interprets the relational content itself, demonstrating robustness to ordering variations and reinforcing the idea that our method is performing global reasoning rather than exploiting data-specific patterns.
6 Complexity Analysis
In this section, we present & comprehensive computational complexity analysis. We first introduce the notation used throughout this section. We examine standard time complexity in classical methods and the offline CPU-side computational complexity in hybrid methods independent of any LLM calls (Section 6.1 We then provide a valuable study specific to LLMs for BN structure discovery, including query complexity (Section 6.2 and scalability (Section 6.3 We present an overview of complexities for our method and compare them to existing methods in Table
Notation Let the BN structure learning problem be defined over d random variables. We denote N as the number of nodes (variables) in the Bayesian network, as the number of directed edges in the true learned DAG, analysis), and l the largest. number of neighbors any variable has al any point during the PC adjacency removal process (i.e., the maximum adjacency size).
the number of iterations (considered as constant in the computational complexity
Table 3: Computational complexity analysis
Algorithm
Category Time Complexity
Query Complexity
Scalability
LLM
O(1)* to O(N2)
PairwisePrompt
LIM
O
BFSPrompt
LLM
O(N)
ChatPC
L.M
O(N2, 2*)
PromptBN (Ours)
LLM
O(1)
PC-Stable
Data
O(N2. 2
HC
Data
O(N2)
GES
Data
O(N2)* to O(N3)
LLM-CD(HC)
Hybrid
O(N2)
O(1) to O(N2) (1)
ReActBN (Ours)
Hybrid
O(N2)
the value represents the best rase depending on an internal LLM's generation step.
6.1 Time Complexity of Offline Computation
Both data-only algorithms and hybrid methods can incur substantial CPU overhead as the number of vari- ables grows. We therefore compare classical data-only methods with the CPU components of hybrid ap- proaches with full derivations, with complete complexity derivations in Appendix C

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PromptBN does not involve observational data and thus has data-processing time complexity. ReActBN performs L..M-uudded refinement with a classical search-and-score procedure, enumerating and evaluat- ing local graph modifications at each iteration. Since each refinement step constructs the same ((N²) neighborhood as HC, the dominant computation per iteration is O(N2). With a bounded number of refine- ment iterations, the overall runtime of ReActBN scales as O(N2). Compared to classical score-based and constraint-based algorithms (Data in Table 3) ReActBN introduces no additional computational overhead and exhibits similar complexity to SOTA hybrid methods such as LLM-CD when instantiated with the same score- based search component (i.e., HC). Nonetheless, ReActBN shows significantly lower query complexity than LLM-CD (HC), as discussed below.
6.2 LLM Query Complexity
LLM calls dominate both computational latency and monetary cost. We therefore study LLM query com- plexity, defined the number of prompt-response interactions, to demonstrate our advantages in these two dimensions. We adopt the notion of query complexity anntlooon to Big-O notation used in traditional time and space complexity analyses to quantify the number of LLM API calls required by each algorithm. This metric applics specifically to LLM-based structural learning methods (LLM and Hybrid in Table 3 The complete query complexily derivations are in Appendix D
PromptBN achieves constant query complexity of (()) by generating the entire candidate DAG in a single LLM invocation, representing the optimal bound for any single-pass generative approach. In contrast, prior LLM-only methods require repeated pairwise or conditional-nndeennee complexity. For example, ChatPC, which replaces the conditional independence (CI) tests in PC-Stable with LLM-based CI judgments, issues one query for each ordered pair of variables and for each conditioning set considered during adjacency removal. There are O(N2) ordered variable pairs, and for each pair the algorithm enumerates up 2* conditioning subsets when the adjacency set has size at most k. The worst-case query complexity of ChatPC is O(N22*). By avoiding this bottleneck, PromptBN maintains competitive structural discovery accuracy while offering substantially lower query costs that would otherwise become prohibitive.
queries, leading to high query
While further integrating LLM reasoning with classical score evaluations, ReActBN maintains query effi- ciency. Since the number of LLM queries equals the number of refinement iterations, which is fixed across datasets following standard Hill Climbing practice, ReActBN achieves query complexity of (1) per iteration. This makes substantially more query-effiiient than existing hybrid methods, which require validation of all candidate edges and thus scale with network size. For example, LLM-CD introduces three-stage prompting strategy. Its metadata derivation stage calls one query per variable, giving cost of O(N). Then, the causal extraction stage invokes the LLM once to extract potential causal relationships. The final causal validation stage checks each extracted directed pair individually. In the worst case, the extraction stage can propose all NNN - 1) ordered pairs, resulting in O(N2) validation queries, resulting in the overall worst-case query complexity O(N2). The advantages of our approach in query complexity over existing LLM-only and hybrid methods further demonstrate its effectiveness.
6.3 Scalability
In practice, datasets vary considerably in size, making scalability to larger networks a critical consideration. We analyze how time and query complexity scale with dataset size and demonstrate both PromptBN and ReActBN are the only methods that (i) prevent LLM query counts from scaling with N and (ii) maintain consistently strong performance across datasets, making them the most scalable solutions among all LLM- based methods considered.
Among LLM-only approaches (LLM), ChatPC is fundamentally unsuitable for moderate- or largo-scale graphs. Its reliance on PC-Stable yields the poorest scalability, with query complexity mirroring the condittionll-nndeeedddeee testing phase: O(N2. in the worst case, becoming exponential when the max- imum adjaceney size grows with N. In contrast, PromptBN achieves optimal scalability by generating the entire DAG in a single LLM inference, resulting in constant query complexity of O(1) and completely avoiding enumeration of conditioning sets or adjacency neighborhoods. PairwisePronpt and BFSPrompt fall
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between these extremes, with quadratic and linear query growth respectively, but they remain significantly less scalable than PromptBN.
7 Factor Analysis
In this section, we study two fundamental factors when combining data with LLMs.
Table 4: Performance under different observational data sample sizes
Asia
Cancer
Child
Insurance
Hailfinder blockchain
covid
disputed3
Category Sample Algorithm
SHD, NHD] SHD NHDD SHDL NHD, SEDDD_SDDINNN NHDISHDL SHDDDHHHDNHDD SHDD NHD SHD NHDJ NHDI
Data
100
HC
8.0 0.125
4.0 0.160
20.0 0.068
54.0 0.067
50.0 0.040
72.0 0.024
13.2 0.094
32.6 0.156
30.8 0.051
Hlbrid
100
PromptBN + IIC
8.0 0.094
0.0 0.000
18.2 0.030
46.6 0.055
43.6
0.028
63.8 0.018
12.2 0.085
33.2 0.156
15.4 0.019
Hybrid
100
ReActNN (Ours)
6.4 0.075
0.0 0.000
18.0 0.028
40.2
0.049
35.4
0.024
75.0
0.023
11.0 0.074
32.8 0.164
12.2 0.015
Data
250
HC
3.0 0.091
10 0,160
17.0 0.018
19.0 0.063
38.0
0.037
76,0 0.025
13.0 0.090
30.0 0.133
11.0 0,078
Hybrid
250
PromptRN IHC
1.0 0.016
0.0 0.000
13.2 0.026
37.6
0.046
32.8
0,026
65.8 0.019
9.6 0.079
28.4 0.140
11.8 0.013
Hybrid
250
ReActBN (Ours)
1.0 0.016
0.0 0.000
12.6 0.028
32.4 0.039
32.2 0.025
69.6
0.022
10.2
0.075
30.8
0.157
9.8 0.012
Table 5: Performance of PromptBN with different LLMs
Asia
Cancer
Child
Insurance
Alarm
Hailfinder
blockchain
covid
disputed3
Model
SHD NHD] SHDI
NHD SHD NHDI SHD NHDI SHDL NHD, SHD, NHDI SHD NHD. SHD1 NHD, SHD,
NHD
o3-pro
0.0 0.000
1.0 0.040
22.2
0.045
37.6
0.041
44.8
0.035
79.6
0.126
12.8
0.085
41.2
0.222
9.8
0.010
63
0.0 0.000
0.6 0.024
21.6
0.044
35.6
0.043
41.8
0,030
76.8
0.157
15.2
0,045
45.0
0.221
15.6
0.018
od-mini
0.0 0.000
22.8
0.050
43.2
0.053
46.8
0.035
81.8
0.155
0.042
0.234
0.024
gpt-4.1
0.0 0.000
0.0 0.000
18.2
0.023
39.6
0.045
37.6
0.027
71.2
0.048
16.0
0.046
46.8
0.241
20.0
0.024
gpt-4o
0.0 0.000
0.0 0.000
22.0
0.040
4622
0.054
32.0
0.021
64.0
0.020
14.4
0.094
44.0
0.229
26.4
0.034
Deepseekee1
0.0 0.000
0.0 0,000
18.2
0.038
48.8
0.037
48.8
0,037
65.4
0.021
13.6
0.044
41.2
0.237
5.2
0.007
Deepseek-e3
0.000
0.0 0.000
21.2
0.046
UnP
UnP
UnP
UnP
65.8
0.021
0.207
0.033
gemini-2.55po
0.0 0.000
0.0 0.000
1841
0.031
33.0
0.038
14.2
0.011
71.8
0.023
9.6
0.067
41.8
0.228
9.4
0.009
gomini-2.0-flash
0.0 0.000
0.0 0.000
21.2
0.045
47.4
0.064
40.4
0.029
54.4
0.017
14.0
0.084
39.8
0.192
28.0
0.038
[1] "UnP" (Unparsable) denotes cases where LLM fails to strictly follow the output formatting requirements specified in the prompt, resulting in a response that cannot be parsed into a valid structure.
Q1: How does the sample size of observational data affect performance? Following widely adopted evaluation protocols (Cui et al.] 2022a), we investigate how access to different amounts of ob- servational data influences the performance of our framework. we compare regimes: 100 and 250 samples. For each setting, we evaluate three variants: (1) HC as a data-only baseline, (2) PH,mptBN+HC, which uses an LLM to initialize HC, and (3) ReActBN, which further leverages LLM reasoning to guide the score-based selection. As shown in Table 4 our proposed ReActBN achieves the best performance under sample sizes of both 100 and 250. For example, on Child, ReActBN achieves an SHD of 12.6, compared to 13.2 for HHC and 17 for HC. Morcover, we note that the performance of conventional data-only methods improves as the amount of data increases, narrowing the gap with ReActBN. This trend is expected, the reliance on LLM priors naturally decreases when sufficient data is available for reliable structure estimation.
approach under two data
Q2: How does the choice of LLM affect structure learning performance? We examine how different language models impact performance by evaluating PromptBN across diverse set of LLMs, in- cluding proprietary models from OpenAl and Google and open-source models from DeepSeek. The tested models range from chat-oriented ones (e.g., DeepSeek-v3, GPTT4o, GPT-4.1) to reasoning-fooee focused mod- cls (o.g., DeepScek-rl, OpenAT o-scries), and general-purpose language models (c.g., 2.5ini-2.5-Pro and We present the results in Table 5 Interestingly, more advanced models with stronger general-purpose reasoning capabilities do not always achieve better performance in structure learning. For
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instance, 03-pro, the strongest reasoning model from OpenAI, outperforms other models most LLM benchmarking datasets. However, 03-pro performs worse than GPT-4o, the oldest model evaluated (o3-pro has an SHD of 44.8, while GPTT-4o achieves 32.0). Similarly, Google's best model, Seminn-22.--Pro, performs worse than Gemini-2.0-Fian on Hailfinder (with SHD 71.8 vs. 54.4).
8 Conclusion
In this work, we present a unified, LLM-centered framework for Bayesian network structure discovery that operates effectively in both data-free and data-aware regimes. The first component, PromptBN, formulates structure learning as single LLM query problem and produces globally consistent DAG with constant query complexity (1). second component, ReActBN, integrates Reason-and-Ace (ReAct)-baane agentic reasoning with offline structural scores, such as BIC, to refine the initial graph in L.I.-as-brain process with also (()) query complexity. Empirical evaluations across ten BN benchmarks demonstrate superior robustness, generalization, and low-data performance compared to state-of-the-ar LLM-only, data-only, and hybrid baselines. Together, these results highlight the potential of LLMs as central mechanism for scalable and principled probabilistic structure discovery.
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A Pseudo Code and Examples
A.1 PromptBN
Pseudo-code of the PromptBN (Phase-1 in our framework) is in Algorithm
Algorithm PromptBN: Bayesian Network Estimation via Prompting
1: Input: Variable metadata table metadata())), where V is the set of variables; LLM model M; retry limit
Output: Bayesian network
Construct meta-prompt P including:
Task dcfinition: instruct the LLM to infer Bayesian network from I Variable schema: include each variable's name, description, and distribution
6:
Output format: specify both node-centric and edge-centric representations
7:
Reasoning protocol: instruct the LLM to reason over variable metadata
8:
Response constraints: require strict formatting, completeness, and acyclicity
retry count
10:
repeat
11:
Oraw f- LLM-Query(I,, P;M)
12:
Parse Oraw to obtain both:
13:
Gnode: node-centric representation
14:
Gedge: edge-centric representation
15:
retry_come +- -retr__ccouui count +1
16: until (consistentt(nnote Gedge) and DAG(Gedde)) or retry_ vetnt_
17: retry_coume N then
18: Raise error halt: Valid output not achieved within N retries.
19: else
20: return Bayesian network G <- Gedge
21: end if
A.2 Input and Output Examples
An example of variable metadata I is shown in Table
Table Example imput: Issia metadata Coaiattaaviiiii (Vssia)
node var
asia var description
distribution
visit to Asia
two-level factor with levels yes and no. two-level factor with levels yes and no. a two-level factor with levels yes and no. a two-level factor with levels yes and no. two-level factor with levels yes and no. two-level factor with levels yes and no.
tub
tuberculosis
smoke
smoking
lung
lung cancer
brone
bronchitis
either
tuberculosis versus lung cancer cancer/ bronchitis
xray
chest X-ray
a two-level factor with levels yes and no.
dysp
dyspnoea
a two-level factor with levels yes and no.
The following is an example an LLM response for Asia dataset, illustrating the defined structure query.
"ba": {
modo_ id": 8, node name": "dyap",
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"parents"; ["bronc", "either"], "deccription: "dyspnoee", "distribation": txo-level
yea "coaditionsl_probee prw.bll__pobbbiiiyy..eeeeee "Pldyeplbrone, either)"
"edges":
"fron"; "bronc",
"to": "dysp", "juatification: "Bronchitia can cause dyspnoea."
"fron"; "either",
*ta": "dyap",
"juttification"; "Either tuberculosia lung cancer dyspncea."
"natwork, sunnary": "The Bayesian Natwork modals the relationshipa batsean visiting Asia, smoking, and respiratory diseaces such tuberculosie, lung cancer, bronchitie. captures these factors influence symptons dyspusea and disgnestic results like chest X-rays."
A.3 ReActBN
Pseudo-code of the ReActBN (Phase-2 in our framework) is in Algorithm
Experiments
B.1 Dataset details
Table 7: Statistics of the datasets used for method validation. covid is short for consequenceCovid.
Dataset
Asia
Cancer
Earthquake
Child Insurance
Alarm
Hailfinder blockchain covid!
disputed3
# Nodes
5
20
27
37
56
15
27
# Edges
8
4
4
25
52
46
66
13
34
34
B.2 Hyperparameters
Table 8: Summary key hyperparameters.
Hyperparameter
Meaning
Value
top-k max iter
the candidate action space size
the maximum number of iterations
20
tabu length
the size of the tabu list the type of structure score
100
scoring
BIC
B.3 Variance Statistics
In all our experiments, we report the mean and standard deviation with 5 valid runs under the same settings. To complement the results in the main article, we report the standard deviation in the following tables to provide clearer view of the stability and robustness of our method:
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Algorithm 2 ReActBN: ReAct-based Score-Aware Search
Require: Initial Bayesian network 9n: variable metadata Z; observational data D
Require: Hyperparameters: top-k k; scoring method S; LLM model M; tabu list length L; maximum
iterations T
Ensure: Refined Bayesian network g*
Go: f
< S(g); TL t 0;
while S do
A x
for all ordered pairs of nodes vu) do FLIP) FLIP) do
5:
for all actions E (ADD, REMOVE,
6:
if a is applicable to (u,u) and applying a keeps the graph acyelic then N + 9 + a(u,v)
7:
if N E TI. then
A AU (a(u,
10:
Sa(u,)) S(N) A a(u,,) c- 8a(u,0)
11:
12:
end if
13:
end if end for
14:
15:
end for
16:
Acand top-k elements of A with the largest A aaju,o)
17:
Build prompl P using (I, 8, Acand) idæ < LLMMQEEY((PPM)
18:
idæ E {-1,0, k k- 1}
19:
if idr 1 then
20:
break
21:
else
22:
a" + Acanalida] Iuew + 9
23:
24:
if not acyclic(Gnew) then continue
25:
26:
end if
27:
TL & TLU [
28:
if [TL] > L. then
29:
remove oldest element from TL
30:
end if
31:
G Gnew
32:
S(g)
33:
end if
34:
t 1
35:
end while
36: return 8* g
Table 9
reports the variance of LLM-only and hybrid experiment, complementing Table 1
Table 10
reports the results of Table in the main article.
Table reports the results of Table in the main article.
C
Detailed Time Complexity Analysis
Classical data-driven structure learning Aggortthss-inchudin score-based approaches such as Hill Climbing (HC) and Greedy Equivalence Search (GES), as well as constraint-bart based approaches such PC-Stable- exhibit distinct computational characteristics.
18

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Published in Transactions on Machine Learning Research (01/2026)
Table 9: LLM-only and Hybrid Evaluation on classical (Asia, Cancer, Child, Insurance, Alarm, Hailfinder) and newer (blockchain, covid, disputed3) Bayesian network datasets with Variance
SHGD
SEID
SHDD
SHUL
XHD.
SEDJ
NBD
SHDN
KHD.
1IM
01000
1966
MINE NO0P
37.-1.33
OUEE L
BLOEE 1 0012
BESPoxuph
0.000
0.0:0000
0,000 +: 0.000
20.4
0.04 = D/00
45.4=122
0.015
0.002
Sealebiliy (GPT-479
22.0
PromptisN (Cuun))
Eyleal
100
LIM-CCD-HC)
0.0
0.000
0.0
0,000
46.5
D:096
G5D
0.000
25.0
0.219
Lighed
110
Parnted herept--ooonnt tas 1.
oul
1
201-11 1:1
BLOAN 1.
0.0033
11.11 N.
GISS A
(4051 a 6.033
Eykeal
100
BESPaougg Datn 0.
0.000
D024
022 04
0,005
2000
$6.4555
0.004
0:0044
0/MI 5.000
42.0
0.000
0.005
lyyeal
230
FromptiNN -BC
DA0RC 2 (1A
2: 5/000
46.6
0
1119
PrergiIX- Chole
Eyreal
001000
Category
Sample Algortthm
Holllinder
Mockehain
coxSd
dippurada
SHD.
NHD
BEDJ
XBD
SHD
XHD.
SEDI
NHID
BESP
20.0::0.8
0.234
0.006
Scabililty (GPT-421
08.0
11M
PromptiiN (Qurna)
26.8 EKT 04255
1.008
15.2
11:100
05.9
0:325
15.6
0.055
Blyora
MMM (TCCCC)
150.0
0005
12.0
(1000)
34.0
D:160
15.0
0.055
Hykeal
Printee Dute.
0114
21.2
0,185
#0011
01.
= D/235
55.6
2.
0,447
Eykeal
PaoggENN IBC
0.000
122
58.2
0.166
330
Eyleal
100
RoAciBN (Ouns
=3
0.003 0.00
11.0
0.074
$0.5
0.64
12.3
0.015
[1] Reported in Babakov et al. (2024).
[2] Reported in Cul et al. [2022b].
1; the sample size is 500 instead of 100 for Hailfinder.
Table 10: Performance under different obscrvational data sample sizes with variance
Catagury # Sumple Algorillum
Asia
Cancer
Child
Alarm
SHM1
NHNN
SHT!
NHN]
SHTI
NHN
SHMI
NHD]
SHT]
NHN
Data
100
HC
8.0
1.125
4.0
0.160
20,0
0.068
54.11
0.367
50,0
0,040
L.M--Da Deta
100
+IC
8.0.0.0
0000000000
0,0.0.0
0000000.000
18.2-1.1
0,000000000
45.6.1.411
0006510,006
41.610.55
0.028_00.001
DiM
100
ReActHN (Ours)
6.422.2
0.07620.026
0.0:0.0
0.000:00000
18.0:1.58
0.028:0.004
40.2244.99
0.040:10.044
35.413.44
0.024:0.001
Tists
250
HC
3.0
0.001
4.0
0.160
17.0
0.048
10.0
0.063
38.0
0.037
LLM-
250
+IC
10100.0
0.016110.000
0.0.1..0
0.00010.000
13:211:00
0026600.000
37.613366
0.086100005
32.83300
0,00
I.MM--ass Datts
250
ReActRN (Ours)
1.00..
0.016+0.000
0.0+0-00
0.000-000000
12.0+134
0.028-0.000
32.4+-416
0.080-00002
32.2+2.49
0.025++0.005
Hailfnder
blockehain
covid
disputeda
Category Sample Algorithm
NIDD
SUII
NIID
SEIDI
NIDI
SUID
NIID
100
HC
72.0
0.024
13.2
0.084
52.6
0.158
30.8
0.051
LLMM- Data
100
HHC
63.810.45
0.01810.000
12221.66
0.080810,022
33.242.00
49
0.150::11008
15.424.34
00019100000
L.M Dati
100
ReAeLLN (Ours)
75.0+:387
0,023-100002
11.0=0.00
0.074#0.006
$2.8++2.49
0.164=00004
12,2+2.86
0.015+H0005
Data
250
ICC
76.0
0.025
13.0
0.000
30.0
0.133
44.0
0.078
LLM
250
+UC
65.810.84
0.01011.001
9.611.34
00079100006
28.411.85
0.1401100004
11.817.36
0001310.006
Dota
250
ReActDN (Ours)
00 6::552
0:022:0.001
10.2:1166
0.015000003
30.8::30
0.157:0.0Ld
9.8:5.85
0.012:1.007
Hill Climbing (HC) as a. local search method, HC evaluates the neighborhood N(9) of each candidate DAG, whose size is ((N2) under standard add, delete, and reverse operators. With i search iterations, the worst-case runtime of HC is therefore O(i N22, consistent with empirical findings that its execution time grows approximately quadratically in N (Liu et al. 2022). Since in practice the number of iterations is usually a constant across datasets, we keep the time complexity of HC at O(N2).
Greedy Equivalence Search (GES) GES performs a two-phase (forward and backward) greedy search over oquivalence classos. Each operator evaluation requires computing score differences for the parent sets affected by the modification. As observed in the FastGES (Ramsey et al.] 2016). the per-phase runtime scales practically as O(N2), though the theoretical upper bound may reach O(N*) when many admissible covered-edge reversals are considered. Thus, the overall time complexity of GES ranges between O(V2) and O(N3) depending on structural density and operator availability.
PC-Stable PC-Stable exhibits the most computationally intensive profile among the classical methods due to its reliance on conditional independence (CT) tests as conditioning-set sizes increase. Its runtime
19

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Published in Transactions on Machine Learning Research (01/2026)
Table 11: Performance of PromptBN with different LLMs with variance
Model
Asia
Caneer
Child
Insurance
Alarm
SHD
NHD
SHD4
NHDI
SHD
NHD
SHD
NHD
SHD
NHD
ol-pro
0.0 0.000 + 0.000
100
0.000 0.000
22.2 4.9
0.045 0.011
3.66
0.011 = 0.005
44.3 4: 5.3
0.033 = 0.004
03
0.0 0.000 + 0.000
0.6 0.6 0.024 + 0.022
2..
0.012
36.6
0.044 = 0.004
41.8 + 5.2
0.031 + 0.004
o4-mini
0.0 0.000 + 0.000
0.0
0.000 = 0.000
2.1
0,050
0.011
4.3
0.033 = 0.007
46.8 8
0.035 +: 0.006
gpt-4.1
0.000 0.0
0.000
0.0 0.0 0.000 1: 0.000
18.2
1.6
0.023 1 0.004
38.6
2.77
0.045 0.002
37.6 3.0
0.027 0.002
spt-4o
0.0
0.000 0.000
0.0 0.0 0.000 0.000
2 0.0
0.040 0.000
46.2
2 0.4
0.054 0.002
32.0 +: 6.1
0.021 +: 0.003
Deepeeeker
0.0
0.000 0.000
0.0 0.000
0.000
18.2
0.013
35.11
0.008 UnP
18.8 :
0.037 = 0.009
D
0.0 0.000
0.000
0.0 0.000
0.000
0.5
0.002
Unl
Unl
UnP
gomini-2.5-pro
0.0 0.0 0.000 0.000
0.0 0.0 0.000
0.000
1.7
0.031
0.011
33.03377
0.038
0.004
14.2+
0.011
0.004
gemini- H.n-
0.0 0.0 0.000 0.000
0.0 = 0.0 0.000 + 0.000
21.2 1.9
0.045
0.006
4.7
0.064
0.006
40.4 1: 3.3
0.029 4: 0.002
Model
Hailfinder
blockchain
consequencoCovid
disputed3
SHD
NHD
SHI]
NHD]
SHDI
NHD]
SHDI
NHDI
od-pro
79.6 à 11.7
0.026 1: 0.004
4.0
0.085 1 0.020
4.
0.228
0.011
9.8 1
0.010
0.002
o3
76.8 6.7
0.025 4: 0.003
15.2 6.5
0.100
0.040
45.0
0.017 0.225
15.6 2.2
0.018 = 0.004
od-mnini
8188 10.8
0.027 + 0.001
13.2
0.092 1 0.001
12.8 (28:330
0.231 2 0.013
19.6
0.021= 0.008
gpt-11
6.8
0.020 + 0.002
16.0 0.0
0.111:0.00
16.8
0,212 = 0.007
2..0
0.025 = 0.006
gpt-4o
64.0 + 0.0
0.020
0.000
14.4 + 0.6 0.107 + 0.004
44.0+ 1.2
0.220 + 0.007
26.4 1.7
0.034 0.003
Deepeekorl
65.4 6.3
0.021 1 0.002
13.6 3.7 0.094 1: 0.026
44.2 = 5.2
0.237
0.024
5.2
0.00T
0.003
Deepeeekv3
UnP
UnP
16.2 1.3
0.118
0.007
41.4
0,207 0.017
28.2 2.1
0.053 0.003
gemini-2.5-puu
71.
0.023
0.002
2.6 0.067 = 0.018
41.8
0,228 0.009
9.4
gemini-22.-flaol
0.017 2.7
0.001
2.7
0.01
39.8
0.192
2.50
0.009
is dominated by the adjacency-removal phase. Following the analysis of (Kalisch & Büthmann 2007 the worst-case number of CI tests grows exponentially with the size of the largest adjacency set. Under our notation, for each unordered variable pair the algorithm may evaluate up to 2$ conditioning subsets when the adjacency set has maximun size k. Since there are O(N2) such pairs, the total number of CT tests is bounded by (NN22). An equivalent classical expression (Kalisch & Bühlmann, 2007 Spirtes et al. 2018 rewrites this as O(N++1) by using the identity Σk= (c) = 2* and bounding the number of nonempty adjacency configurations accordingly. When h grows with N-as
as in dense graphs where k = ((N)) PPC Stable becomes exponential in the number of variables, consistent with empirical observations of its rapidly increasing runtime in high-dimensional scttings (Ha et al. 2014).
D Detailed Query Complexity Analysis
ChatPC which replaces the conditional independence (CT) tests in PC-Stable with LLM-based CI judg- ments, issues one query for each ordered pair of variables and for each conditioning set considered during adjacency removal. Using our notation, there are O(N2) ordered variable pairs, and for each pair the al- gorithm enumerates up to 2* conditioning subsets when the adjacency set has size at most k, where k denotes the maximum adjacency size encountered during the search. Thus, the worst-casc qucry complexity of ChatPC is O(N2*),, which can be expressed as (N*+1). This reduces to a polynomial bound when k is small (sparse graphs) but becomes exponential N when le = O(N) (dense graphs).
PairwisePrompt evaluates all N(N-1) possible ordered edges using a single LLM query per pair, yielding a query complexity of O(N2). BFSPrompt reduces this cost by exploring the graph in a breadth-firg manner and issuing exactly one query per node expansion, giving a total of ((N) queries.
Scalability and PromptBN requires only a small number of LLM invocations that do not scale with N. Scalability proceeds in three stages: (1) a facilitator LLM generates M expert personas using one query; (2) each expert receives two prompts to propose and summarize candidate causal relations, contributing 2M additional queries; and (3) optional decycling prompts are issued only when an expert outputs a directed cycle. In the theoretical worst case, resolving all possible conflicts would require ((2) additional queries,
20

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Published in Transactions on Machine Learning Research (01/2026)
since there are most [(N - 1) potentially inconsistent directed pairs. Hence, the algorithm's worst-case query complexity is O(N2); however, with fixed M and empirically rare cycles, the effective query cost behaves as (1) with respect to N.
Pairwiseeromppt--aaae and BESPrompit-Dataa preserve the same interaction structure as in Pair- wisePrompt and BFSPrompt, and additionally incorporate observations into their prompts. Consequently, the query complexities of remain O(N2) and O(N), respectively.
LLM-CD introduces a three-stage prompting strategy. The metadata derivation stage calls one query per variable, giving a cost of O(N). The causal extraction stage invokes the LLM once, using all enriched meta- data, to extract the potential causal relationships. The final causal validation stage checks each extracted directed pair individually; in the worst case, the extraction stage may propose all N NNN - -1) ordered pairs, resulting in O(N2) validation queries. Thus, the overall worst-case query complexity of LLM-CD is O(N²), dominated by the validation stage.
ReActBN alternates between LLM-guided refinement decisions and classical score evaluations. The num-- ber of LLM queries equals the number of refinement iterations prior convergence. Although we denote the iteration count by i, in practice we fix it across datasets, mirroring the common setup in Hill Climbing. As result, the practical query complexity of ReActBN behaves O(1), making it substantially more query- efficient than existing hybrid methods such as LLM-CD, which require validation of all candidate edges. This allows ReActBN to leverage observational data for refinement while avoiding the large query overheads charactoristic of prior LLM+Data techniques.
21


Reviews:
[{"review_id": "BcuQjfrzN8", "reviewer": "Reviewer_UvjX", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a two‑phase LLM‑centered framework for Bayesian network (BN) structure discovery from variable metadata and, optionally, observational data. Specifically, in the data-free case, the paper introduces PromptBN to query LLMs with metadata, and when observational data are available, the paper introduces ReActBN, which integrates the ReAct reasoning paradigm with structure scores such as the Bayesian Information Criterion (BIC) for iterative refinement.\n\nExperiments cover seven classic bnlearn networks and three newer bnRep networks. PromptBN is compared to prior LLM‑based methods and traditional data‑driven structure learners. ReActBN is compared to these data‑driven baselines and to a hybrid PromptBN+HC initialization scheme. Experiments show that the proposed method achieves strong performance. In particular, it accurately recovers exact structures on small networks and maintains high accuracy on medium-sized graphs, even with limited data.", "claims_and_evidence_yn": "No", "claims_explanation": "There are two major concerns:\n1. Training-data contamination: Many classic BN benchmarks (Asia, Cancer, Alarm, Child, Insurance, Hailfinder, Earthquake) and their graphs have been widely used and documented for years; they are very likely present (at least partially) in the training corpora of modern LLMs. The strong performance of PromptBN in the “0‑sample” setting, especially on well‑known benchmarks (e.g., exact recovery on Asia and Cancer), could plausibly be explained by memorization rather than genuine reasoning from metadata.\n\n2.  Complexity and scalability claims are somewhat overstated: The paper repeatedly claims O(1) query complexity for PromptBN and ReActBN. This is true when the number of LLM calls is a function of the number of nodes N, but: (1) ReActBN performs O(N²) offline work per iteration by enumerating all possible local moves for all ordered node pairs, computing scores, and selecting top‑k. (2) The number of iterations T is a hyperparameter. So total offline complexity is O(T·N²), which can be substantial for large graphs and is not analyzed empirically.\n\nThus, “O(1) query complexity” may be technically correct but somewhat misleading: wall‑clock cost scales significantly with N, and there is no runtime or token‑count analysis. A more nuanced treatment of both computational and monetary cost (tokens/API calls) would be desirable.", "audience_yn": "Yes", "audience_explanation": "The paper clearly targets BN structure discovery from metadata, and makes an explicit distinction between data‑free and data‑aware settings. The two‑phase design (PromptBN + ReActBN) is conceptually simple and well‑motivated by limitations of both pure LLM elicitation and pure data‑driven search. The experiments are comprehensive and demonstrate the effectiveness of the proposed two methods.", "requested_changes": "1. Provide a more explicit and nuanced discussion of the risk of data contamination to validity, possibly with simple tests such as: 1) masking or perturbing variable names and descriptions, 2) scrambling node labels and checking whether performance degrades.\n\n\n\n2. Provide a more in-depth analysis of the complexity of the proposed methods. A more nuanced treatment of both computational and monetary cost (tokens/API calls) would be desirable.", "broader_impact_concerns": "No broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "C6oW07QgW9", "reviewer": "Reviewer_zSLz", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "**Summary.**\n\nThe paper proposes a two‑phase, LLM‑centric framework for Bayesian Network (BN) structure discovery:\n\n- **Phase 1 (PromptBN):** single‑call meta‑prompting that elicits a full DAG from variable metadata, with dual output formats (parents‑per‑node and edge list) and acyclicity/consistency checks; regeneration on failure.\n- **Phase 2 (ReActBN):** an agentic refinement loop that exhaustively enumerates local neighbors (add/remove/flip), scores them with BIC, surfaces top‑k candidates, and lets the LLM pick an action (or terminate) under a tabu regime.\n\nExperiments on 7 classic bnlearn networks and 3 bnRep networks compare PromptBN/ReActBN with existing LLM-based approaches and classical methods (HC, PC-Stable, GES, rIeB, rBFchi2) using SHD/NHD, and discuss LLM query complexity. Ablations vary in sample size and LLM family.\n\n**Strengths.**\n\nS1: The PromptBN / ReActBN framework is well-structured and clearly presented, with both high-level intuition (Figure 1) and pseudocode (Algorithms 1 and 2). This makes it relatively easy for readers to understand how the system works end-to-end.\n\nS2: On several benchmark networks, PromptBN alone recovers reasonable structures, and using PromptBN as an initializer or ReActBN as a refinement step often improves over plain HC at 100 samples, especially on Child, Insurance, Alarm, and two of the three newer bnRep networks.\n\nS3: Ablation studies (sample size and LLM choice) are helpful and generally consistent with intuition.\n\n**Weaknesses.** \n\nW1: Algorithm 2 accepts *any* LLM-selected move from the top-k candidates, even when the score difference Δ is negative. There is no explicit constraint “only accept if score improves” and no tracking of the best-scoring graph seen so far; the algorithm simply returns the last graph visited.\n\nW2:  There is no score‑only top‑k (no‑LLM) or top‑k random control using the same enumeration/BIC/tabu for ReActBN, so the incremental value of the LLM’s choice in Phase 2 is unclear. \n\nW3: NHD is defined using a denominator of N² while described as “normalized by the total number of possible edges.” For directed graphs without self-loops, this is N(N−1); clarifying that they normalize by N² adjacency entries (including diagonal) or adjusting the description would improve clarity. \n\nW4: The BIC scoring setup is not fully specified (e.g., discrete vs continuous variables, parameter estimation method, implementation details), which could matter for reproducibility and for interpreting SHD/NHD differences.\n\nW5: While the prompt structure is described, specific details required for reproduction are missing or risky. The paper uses specific model snapshots (e.g., `o3-2025-04-16`), but code is only promised \"upon publication.\" Given the sensitivity of LLM-based evaluations to prompt wording and temperature settings, this hinders reproducibility.", "claims_and_evidence_yn": "No", "claims_explanation": "C1: The efficiency discussion and Table 5 focus on LLM query complexity (O(1) for PromptBN and ReActBN) but do not analyze the computational cost of repeated neighbor enumeration and BIC rescoring, which is O(N²) neighbors per iteration times the number of iterations. In practice, this could become expensive for larger graphs, even if the number of LLM calls is small. Yet the paper presents the method as efficient without discussing CPU runtime.\n\nC2: Most classic BN benchmarks used here are textbook networks that LLMs might have seen during pre-training; the “data-free” story, therefore, needs a more explicit discussion of pre-training overlap vs. genuine reasoning.\n\nC3: Table 3 mixes numbers reproduced here (PC‑Stable/HC at 100 samples with pgmpy *defaults*) with results imported from other papers at unknown or different sample sizes (e.g., GES; several LLM+Data entries). This weakens head‑to‑head conclusions and may overstate gains.", "audience_yn": "Yes", "audience_explanation": "The paper sits at the intersection of Bayesian networks, causal/structure discovery, and LLM-based reasoning/agents, all topics of active interest at TMLR. A method that can generate BNs from metadata alone and refine them with a ReAct-style loop is relevant to readers working on hybrid neuro-symbolic systems, LLM-guided search, and data-efficient structure learning.", "requested_changes": "Solve or clarify W1-5, C1-3.", "broader_impact_concerns": "I do not see major ethical red flags requiring a long Broader Impact section, but one point is worth emphasizing: the paper promotes LLM-assisted discovery of probabilistic/causal structures. If deployed in high-stakes domains (e.g., medicine, policy, finance) without strong domain-expert validation, there is a risk of over-trusting partially-validated structures. A short note reminding practitioners that these BNs should be treated as hypotheses rather than ground truth would be helpful.", "additional_comments": "This is a timely and clearly written paper with an appealing high-level idea: use LLMs both to propose an initial BN from metadata and to guide a score-based refinement loop. The experiments are reasonably extensive, and the ablations are useful.\n\nHowever, I have substantial concerns about:\n\n- The clarity and correctness of the ReActBN search procedure as currently specified.\n- The lack of a score-only / random control to isolate the LLM’s contribution in ReActBN.\n- The limited analysis of computational cost beyond LLM query counts.\n- The potential for pre-training overlap in “data-free” experiments. \n- The mixing of in-house vs imported baselines with different conditions.", "extra_scores": {}, "extra_text": {}}, {"review_id": "rYc9CEOQ2F", "reviewer": "Reviewer_5trM", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "1. A framework for Bayesian structure learning is proposed (PromptBN and ReActBN) that leverages Large Language Models to efficiently discover Bayesian Networks.\n\n2. The iterative procedure of ReActBN, where an LLM agent can modify the graph based on scores computed from data, is a novel idea.\n\n3. The experimental results presented in the paper are promising. However, the extent of the evaluation is quite limited.", "claims_and_evidence_yn": "No", "claims_explanation": "1. **Incomplete Comparative Analysis:** Table 3 would be a good basis for experimental evaluation, but the overwhelming lack of results greatly weakens the validity of the results. For example, the purely LLM-based comparisons only show results on 2 or 3 of the 6 datasets, and many results are missing for the other methods as well. As I understand, the authors did not run these experiments themselves and only reported what was done in the original papers, correct? While this is generally not a problem, it leads to too many empty fields here, and I recommend running more of these experiments yourself.\n\n2. **Evaluation on Lesser-Known Datasets:** I really like that the paper includes three lesser-known datasets (blockchain, covid, disputed3), especially since evaluating LLMs on famous datasets might always be biased, as they were likely to appear in their training data. It would be great if all experiments conducted for Table 3 could also be run on these datasets.\n\n3. **Scalability:** A more extensive discussion on the performance of PromtBN and ReActBN on larger datasets and how well they scale would be helpful. For example, sorting the datasets by their variable size could also help illustrate how well the framework scales to larger problems.", "audience_yn": "Yes", "audience_explanation": "Causal discovery is a problem many people are interested in. Apart from purely data-based and purely LLM-based methods, hybrid methods that aim to combine the strengths of both categories have begun to emerge. This paper proposes one such method that is novel and should be of some interest to the causal discovery community.", "requested_changes": "1. **Experimental Results.** This is crucial for readers to assess the potential of this framework. Providing the missing results in Table 3 would greatly strengthen the experimental evaluation. Additionally, including the three less well-known datasets there (blockchain, covid, disputed3) would nicely complement these results.\n2. **Questions.** Please respond to my questions listed in \"Additional Comments\" and consider clarifying these points in the manuscript as well. These should not be major points, but I think they could lead to changes that can improve the paper.", "broader_impact_concerns": "None", "additional_comments": "**Questions**\n\n1. **Variable Descriptions:** Were all variable descriptions human-made? As the method relies on an LLM, have the authors considered querying the LLM to *generate* these descriptions as well, and how might that impact performance?\n2. **Source of Variance:** The standard deviations reported in the appendix are noted. Is this variance caused *only* by the LLM's stochasticity, or are there other sources of randomness in the experiments?\n3. **Score Computation and Alternatives:** Could the authors provide a  more detailed breakdown of how the score in ReActBN is computed (is it always BIC)? Could other scoring functions or even more different approaches be used as well?\n4. **Invalid Runs:** The paper mentions some runs were \"invalid.\" Could the authors please clarify how often invalid runs occured and the specific reason for their invalidity? For instance, did they output cyclic graphs or fail to parse?\n5. **Choice of LLMs:** What was the rationale for using different LLMs for PromptBN (GPT-4) and ReActBN (GPT-3.5)?\n6. **ChatPC Complexity:** The $O(N^3)$ query complexity for ChatPC seems to assume only one query per conditioning set. Given that conditioning sets themselves grow worse than linearly, could the authors clarify how the $O(N^3)$ complexity is derived?\n7. **LLM-CD Complexity:** Similarly, could the authors please provide a brief explanation or derivation for how the $O(N^2)$ worst-case complexity for LLM-CD is determined?\n8. **Reasoning Model Performance:** The paper notes that reasoning-focused models performed worse. Do the authors have a hypothesis as to why this might be, or is this attributed primarily to experimental noise?\n\n**Minor Suggestions**\n\n- **\"Ablation Study\" Naming:** Please consider renaming the \"Ablation Study.\" These experiments are valuable, but they don't seem to fit the standard definition of an ablation (i.e., removing a method component).\n- **Table 7 Highlighting:** Please add boldface highlighting to Table 7 to indicate the best-performing methods, consistent with the other tables.\n- **Figure 1 Font Size:** The text in Figure 1 is quite small. Please increase the font size for readability.\n- **Citation Formatting:** There are numerous `citep`/`cite` formatting errors (e.g., 4.1 \"Zhang et al. (2023)\" and 5.1 \"Following (Khatibi et al., 2024),\"). Please perform a thorough check of all citations.\n- **Section 5 Title:** The section title \"Experiment\" (singular) does not sound great. Please consider changing it to \"Experiments\" or \"Experimental Evaluation.\"\n- **Section Order:** The last sentence before 5.1. previews the sections out of order (mentions 5.6 before 5.5). Please correct this to match the paper's structure.\n- **Table 3 Scaling:** The \"scaled by x10^-3\" in Table 3 is slightly confusing, as the text uses the decimal values. Please consider putting the full decimal values (e.g., 0.068) in the table for clarity and consistency.\n- **Text/Table Mismatch:** The HC value for the CHILD dataset in the text (0.065) does not match the value in Table 3 (0.068). Please check and correct this.\n- **Algorithm Line Numbers:** The line numbering in Algorithms 1 and 2 appears to be broken or rendered incorrectly.\n- **Typo:** There is a typo in the text \"250 ̆1000\" (the \" ̆\" character).", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)