Bayesian Networks

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A Bayesian network uses a directed acyclic graph and local conditional probability models to represent how variables depend on one another, supporting tasks such as diagnosis, prediction, and information gathering. Researchers learn graphs from data using score-based and constraint-based search, often adding expert knowledge or priors, while inference methods range from exact algorithms to sampling and neural approximations. Applications span causal discovery, clinical modeling, time series, and synthetic data generation, with growing work combining networks with language models and differentiable mechanisms. Key challenges include scaling inference and structure search, distinguishing causal direction from observational association, and handling latent confounding, sparse data, and uncertain or imperfect prior knowledge.

Latest papers

47 papers6 papers in the last four weeks, down 25% on the four weeks before. 0.0% of all new papers.

May 26, 2026

Iterative Causal Discovery: Per-Edge Impossibility Certificates, Tier-Aware Oracle Queries, and the $1+K$ Lower Bound

Causal-discovery algorithms return a directed graph, yet provide no principled means of distinguishing edge directions identified by the data from those assigned without an identifying assumption. Under the standard Markov and faithfulness conditions, the observational distribution identifies only a Markov equivalence class; orientations within that class are not determined by the joint distribution and cannot be recovered from additional samples alone, but require either a functional restriction or an intervention. We introduce a protocol for observational causal discovery on continuous data that attaches to each candidate edge a discrete impossibility certificate: a RESOLVED code records the identifiability theorem under which the direction was committed, while an IMPOSSIBLE code records the failure mode together with the specific question a domain expert must answer to resolve it. The bivariate cascade is extended with five gated identifiability tiers LSNM, IGCI, Stein, MDL, and PEIT that abstain when their precondition test rejects. Two oracle primitives, the meta-hub query and the node-children query, jointly establish an upper bound of

1+K

expert interactions sufficient to recover any DAG, where

K

denotes the number of non-leaf vertices. Under an ideal-oracle assumption, the bound is met exactly on the asia, sachs, child, and alarm benchmarks.

Eichi Uehara

aflo, Inc., Tokyo, Japan

Bayesian Networks Causal Discovery Causal Structure Learning Causal Identifiability

May 15, 2026

Bayesian Networks for Path-Based Sensors: Gathering Information and Path Planning in Communication Denied Environments

A "path-based sensor" produces a single observation along a continuous path. For example, a boolean path-based sensor returns a single "1" if an event of interest is detected at any point along the path and a "0" otherwise. Notably, a "1" provides no direct information about where along the path the event(s) may have occurred. Previous work has demonstrated that observations from multiple path-based sensors can be fused to create a Bayesian belief map over the spatial locations of the underlying event or phenomenon. Moreover, path planning can employ Shannon information theory to accelerate the rate of convergence of the belief map. In this paper, we present a new method to update the belief map based on a path-based sensor observation, and then plan paths to increase information gain. In contrast to prior work that approximates the posterior by averaging over the alternative event histories, we introduce a Bayesian Network (BN) formulation that models the probabilistic relationships between the latent variables and path-based sensor measurements, enabling a more principled Bayesian belief update. We consider static hazard detection in a communication-denied environment as a representative problem setting. The event of a robot returning from its path corresponds to a path-based hazard sensor reading of "0" (hazard not detected), while a robot failing to return corresponds to a reading of "1" (hazard detected). We consider false positives and false negatives. We find that the new method leads to quicker convergence of the belief map than prior work in both single- and multi-robot cases.

Alkesh K. Srivastava, George P. Kontoudis, Donald Sofge +1

University of Maryland, College Park, MD, US. · Temple University, Philadelphia, PA, US. · Colorado School of Mines, Golden, CO, US. +1

Bayesian Networks Informative Path Planning Motion Planning

May 11, 2026

ANCHOR: Abductive Network Construction with Hierarchical Orchestration for Reliable Probability Inference in Large Language Models

A central challenge in large-scale decision-making under incomplete information is estimating reliable probabilities. Recent approaches use Large Language Models (LLMs) to generate explanatory factors and coarse-grained probability estimates, which are then refined by a Naïve Bayes model over factor combinations. However, sparse factor spaces often yield

unknown'' predictions, while expanding factors increases noise and spurious correlations, weakening conditional independence and degrading reliability. To address these limitations, we propose Anchor, an aggregated Bayesian inference framework over a hierarchical factor space. It constructs dense factor hierarchies through iterative generation and clustering, maps contexts via hierarchical retrieval and refinement, and augments Naïve Bayes with a Causal Bayesian Network to model latent factor dependencies. Experiments show that Anchor markedly reduces

unknown'' predictions and produces more reliable probability estimates than direct LLM baselines, achieving state-of-the-art performance while significantly reducing time and token overhead.

Wentao Qiu, Guanran Luo, Zhongquan Jian +3

School of Informatics, Xiamen University, Xiamen, China · School of Computer and Data Science, Minjiang University, Fuzhou, China · School of Film, Xiamen University, Xiamen, China +2

Abductive Reasoning Bayesian Networks Bayesian Inference Probabilistic Graphical Models LLM Reliability LLM Uncertainty Estimation

May 11, 2026

Free Energy Manifold: Score-Based Inference for Hybrid Bayesian Networks

We introduce the Free Energy Manifold (FEM), a score-trained conditional energy model specialized for inference in hybrid Bayesian networks with discrete and continuous variables. FEM represents each conditional factor as an energy landscape over learned discrete-parent embeddings and continuous observations, enabling posterior evaluation, generative sampling, and compositional inference across multiple continuous leaves by energy addition under conditional independence. A central finding is the mode-bridge artifact: standard conditional energy models can create low-energy ridges between separated modes of the same class, producing overconfident posteriors at off-data interior points. We analyze this failure and propose valley regularization, an off-data calibration term that restores near-uniform posteriors in such regions while preserving in-data fit. Across synthetic multimodal hybrid-BN benchmarks, FEM substantially reduces KL divergence relative to classical baselines and a vanilla conditional EBM, including large gains at mode-bridge midpoint queries and in multi-leaf evidence composition. We also evaluate high-cardinality discrete-parent settings and a UCI Breast Cancer sanity check, showing that FEM is most useful when multimodal or compositional Bayesian-network inference is required, while discriminative classifiers remain preferable for closed-world classification tasks.

Cheol Young Park, Shou Matsumoto

ATOS Co., Ltd., 273, Digital-ro, Guro-gu, Seoul, 08381, Republic of Korea · C5I Center, George Mason University, Fairfax, VA 22030, USA

Bayesian Networks Energy-Based Models Conditional Distribution Estimation Score-Based Generative Modeling Uncertainty Calibration

May 5, 2026

Exact and Approximate Algorithms for Polytree Learning

Polytrees are a subclass of Bayesian networks that seek to capture the conditional dependencies between a set of

n

variables as a directed forest and are motivated by their more efficient inference and improved interpretability. Since the problem of learning the best polytree is NP-hard, we study which restrictions make it more tractable by considering for example in-degree bounds, properties of score functions measuring the quality of a polytree, and approximation algorithms. We devise an algorithm that finds the optimal polytree in time

O((2+ε)^n)

for arbitrarily small

ε> 0

and any constant in-degree bound

k

, improving over the fastest previously known algorithm of time complexity

O(3^n)

. We further give polynomial-time algorithms for finding a polytree whose score is within a factor of

k

from the optimal one for arbitrary scores and a factor of

2

for additive ones. Many of the results are complemented by (nearly) tight lower bounds for either the time complexity or the approximation factors.

Juha Harviainen, Frank Sommer, Manuel Sorge

Department of Computer Science, University of Helsinki, Helsinki, Finland · Institute of Computer Science, Friedrich-Schiller Universität Jena, Jena, Germany · Institute of Logic and Computation, TU Wien, Vienna, Austria

Graph Structure Learning Bayesian Networks Probabilistic Graphical Models Approximation Algorithms Bayesian Network Structure Learning

Apr 15, 2026

Time series causal discovery with variable lags

Causal Bayesian Networks (CBNs) are a powerful tool for reasoning under uncertainty about complex real-world problems. Such problems evolve over time, responding to external shocks as they occur. To support decision-making, CBNs require a cause-and-effect map of the variables under consideration, known as the network's structure. Learning the graphical structure of a causal model from data remains challenging; learning it from time-series data is even harder because dependencies may arise at different time lags. Existing time-series causal discovery methods often assume a fixed lag window and do not explicitly optimise edge-specific lags. We propose a Tabu-based structure learning algorithm that searches for a time-ordered directed structure (i.e., where every edge respects time) while allowing edge-specific lags up to a specified maximum lag. The approach uses a decomposable BIC-based score with node-specific effective sample sizes and an explicit lag-length penalty encouraging parsimonious delay assignments while preserving efficient local score updates. We provide theoretical guarantees of validity and local optimality, and we also describe a parallel implementation for improved scalability. In simulations, the method recovered graph structure competitively and estimated lags accurately when true adjacencies were recovered. On a real-world UK COVID-19 policy dataset, the learnt structure was dominated by short delays while retaining a substantial minority of longer-lag dependencies, consistent with delayed behavioural and epidemiological effects.

Bruno Petrungaro, Anthony C. Constantinou

Bayesian Artificial Intelligence research lab, MInDS research group, Queen Mary University of London, Mile End Road, London, E1 4NS, UK.

Time-Series Causal Discovery Causal Structure Learning Bayesian Network Structure Learning Bayesian Networks

Apr 1, 2026

Transfer learning for nonparametric Bayesian networks

This paper introduces two transfer learning methodologies for estimating nonparametric Bayesian networks under scarce data. We propose two algorithms, a constraint-based structure learning method, called PC-stable-transfer learning (PCS-TL), and a score-based method, called hill climbing transfer learning (HC-TL). We also define particular metrics to tackle the negative transfer problem in each of them, a situation in which transfer learning has a negative impact on the model's performance. Then, for the parameters, we propose a log-linear pooling approach. For the evaluation, we learn kernel density estimation Bayesian networks, a type of nonparametric Bayesian network, and compare their transfer learning performance with the models alone. To do so, we sample data from small, medium and large-sized synthetic networks and datasets from the UCI Machine Learning repository. Then, we add noise and modifications to these datasets to test their ability to avoid negative transfer. To conclude, we perform a Friedman test with a Bergmann-Hommel post-hoc analysis to show statistical proof of the enhanced experimental behavior of our methods. Thus, PCS-TL and HC-TL demonstrate to be reliable algorithms for improving the learning performance of a nonparametric Bayesian network with scarce data, which in real industrial environments implies a reduction in the required time to deploy the network.

Rafael Sojo, Pedro Larrañaga, Concha Bielza

Aingura IIoT, Paseo Mikeletegui 43, 20009 Donostia-San Sebastián, Gipuzkoa, Spain · Universidad Politécnica de Madrid, Departamento de Inteligencia Artificial, 28660 Boadilla del Monte, Madrid, Spain

Bayesian Network Structure Learning Transfer Learning Bayesian Networks Negative Transfer

Mar 28, 2026

On the Relationship between Bayesian Networks and Probabilistic Structural Causal Models

In this paper, the relationship between probabilistic graphical models, in particular Bayesian networks, and causal diagrams, also called structural causal models, is studied. Structural causal models are deterministic models, based on structural equations or functions, that can be provided with uncertainty by adding independent, unobserved random variables to the models, equipped with probability distributions. One question that arises is whether a Bayesian network that has obtained from expert knowledge or learnt from data can be mapped to a probabilistic structural causal model, and whether or not this has consequences for the network structure and probability distribution. We show that linear algebra and linear programming offer key methods for the transformation, and examine properties for the existence and uniqueness of solutions based on dimensions of the probabilistic structural model. Finally, we examine in what way the semantics of the models is affected by this transformation. Keywords: Causality, probabilistic structural causal models, Bayesian networks, linear algebra, experimental software.

Peter J. F. Lucas, Eleonora Zullo, Fabio Stella

ICIS, Radboud University, Nijmegen, the Netherlands · Dipartimento di Informatica, Sistemistica e comunicazion University of Milano-Bicocca, Milan, Italy

Structural Causal Models Bayesian Networks Probabilistic Graphical Models

Mar 21, 2026

Neural Autoregressive Flows for Markov Boundary Learning

Recovering Markov boundary -- the minimal set of variables that maximizes predictive performance for a response variable -- is crucial in many applications. While recent advances improve upon traditional constraint-based techniques by scoring local causal structures, they still rely on nonparametric estimators and heuristic searches, lacking theoretical guarantees for reliability. This paper investigates a framework for efficient Markov boundary discovery by integrating conditional entropy from information theory as a scoring criterion. We design a novel masked autoregressive network to capture complex dependencies. A parallelizable greedy search strategy in polynomial time is proposed, supported by analytical evidence. We also discuss how initializing a graph with learned Markov boundaries accelerates the convergence of causal discovery. Comprehensive evaluations on real-world and synthetic datasets demonstrate the scalability and superior performance of our method in both Markov boundary discovery and causal discovery tasks.

Khoa Nguyen, Bao Duong, Viet Huynh +1

Deakin Applied Artificial Intelligence Initiative, Geelong, Australia · Edith Cowan University, Perth, Australia

Causal Structure Learning Causal Discovery Bayesian Networks

Mar 17, 2026

Transformers are Bayesian Networks

Transformers are the dominant architecture in AI, yet why they work remains poorly understood. This paper offers a precise answer: a transformer is a Bayesian network. We establish this in five ways. First, we prove that every sigmoid transformer with any weights implements weighted loopy belief propagation on its implicit factor graph. One layer is one round of BP. This holds for any weights -- trained, random, or constructed. Formally verified against standard mathematical axioms. Second, we give a constructive proof that a transformer can implement exact belief propagation on any declared knowledge base. On knowledge bases without circular dependencies this yields provably correct probability estimates at every node. Formally verified against standard mathematical axioms. Third, we prove uniqueness: a sigmoid transformer that produces exact posteriors necessarily has BP weights. There is no other path through the sigmoid architecture to exact posteriors. Formally verified against standard mathematical axioms. Fourth, we delineate the AND/OR boolean structure of the transformer layer: attention is AND, the FFN is OR, and their strict alternation is Pearl's gather/update algorithm exactly. Fifth, we confirm all formal results experimentally, corroborating the Bayesian network characterization in practice. We also establish the practical viability of loopy belief propagation despite the current lack of a theoretical convergence guarantee. We further prove that verifiable inference requires a finite concept space. Any finite verification procedure can distinguish at most finitely many concepts. Without grounding, correctness is not defined. Hallucination is not a bug that scaling can fix. It is the structural consequence of operating without concepts. Formally verified against standard mathematical axioms.

Gregory Coppola

Belief Propagation Transformer Bayesian Networks Transformer Interpretability

Mar 16, 2026

Estimating Staged Event Tree Models via Hierarchical Clustering on the Simplex

Staged tree models enhance Bayesian networks by incorporating context-specific dependencies through a stage-based structure. In this study, we present a new framework for estimating staged trees using hierarchical clustering on the probability simplex, utilizing simplex basesd divergences. We conduct a thorough evaluation of several distance and divergence metrics including Total Variation, Hellinger, Fisher, and Kaniadakis; alongside various linkage methods such as Ward.D2, average, complete, and McQuitty. We conducted the simulation experiments that reveals Total Variation, especially when combined with Ward.D2 linkage, consistently produces staged trees with better model fit, structure recovery, and computational efficiency. We assess performance by utilizing relative Bayesian Information Criterion (BIC), and Hamming distance. Our findings indicate that although Backward Hill Climbing (BHC) delivers competitive outcomes, it incurs a significantly higher computational cost. On the other, Total Variation divergence with Ward.D2 linkage, achieves similar performance while providing significantly better computational efficiency, making it a more viable option for large-scale or time sensitive tasks.

Muhammad Shoaib, Eva Riccomagno, Manuele Leonelli +1

University of Genova, Department of Mathematics, Genova, 16146, Italy · IE University, School of Science and Technology, Madrid, Spain · Universitat de València, Department of Statistics and Operational Research, València, , Spain

Bayesian Networks Hierarchical Clustering

Mar 15, 2026

Scaling the Explanation of Multi-Class Bayesian Network Classifiers

We propose a new algorithm for compiling Bayesian network classifier (BNC) into class formulas. Class formulas are logical formulas that represent a classifier's input-output behavior, and are crucial in the recent line of work that uses logical reasoning to explain the decisions made by classifiers. Compared to prior work on compiling class formulas of BNCs, our proposed algorithm is not restricted to binary classifiers, shows significant improvement in compilation time, and outputs class formulas as negation normal form (NNF) circuits that are OR-decomposable, which is an important property when computing explanations of classifiers.

Yaofang Zhang, Adnan Darwiche

Department of Computer Science University of California, Los Angeles

Explainable Artificial Intelligence Bayesian Networks Multiclass Classification

Mar 10, 2026

Learning Bayesian and Markov Networks with an Unreliable Oracle

We study constraint-based structure learning of Markov networks and Bayesian networks in the presence of an unreliable conditional independence oracle that makes at most a bounded number of errors. For Markov networks, we observe that a low maximum number of vertex-wise disjoint paths implies that the structure is uniquely identifiable even if the number of errors is (moderately) exponential in the number of vertices. For Bayesian networks, however, we prove that one cannot tolerate any errors to always identify the structure even when many commonly used graph parameters like treewidth are bounded. Finally, we give algorithms for structure learning when the structure is uniquely identifiable.

Juha Harviainen, Pekka Parviainen, Vidya Sagar Sharma

University of Helsinki, Finland · University of Bergen, Norway · Indian Institute of Technology Madras, India

Graph Structure Learning Bayesian Network Structure Learning Probabilistic Graphical Models Bayesian Networks Markov Random Fields

Mar 9, 2026

Quantization of Ricci Curvature in Information Geometry

In 2004, while studying the information geometry of binary Bayesian networks (bitnets), the author conjectured that the volume-averaged Ricci scalar <R> computed with respect to the Fisher information metric is universally quantized to positive half-integers: <R> in (1/2)Z. This paper resolves the conjecture after 20 years. We prove it for tree-structured and complete-graph bitnets via a universal Beta function cancellation mechanism, and disprove it in general by exhibiting explicit loop counterexamples. We extend the program to Gaussian DAG networks, where a sign dichotomy holds: discrete bitnets have positive curvature, while Gaussian networks form solvable Lie groups with negative curvature.

Carlos C. Rodriguez

Department of Mathematics and Statistics, University at Albany, SUNY

Information Geometry Riemannian Geometry Bayesian Networks

Mar 5, 2026

Learning Causal Structure of Time Series using Best Order Score Search

Causal structure learning from observational data is central to many scientific and policy domains, but the time series setting common to many disciplines poses several challenges due to temporal dependence. In this paper we focus on score-based causal discovery for multivariate time series and introduce TS-BOSS, a time series extension of the recently proposed Best Order Score Search (BOSS) (Andrews et al. 2023). TS-BOSS performs a permutation-based search over dynamic Bayesian network structures while leveraging grow-shrink trees to cache intermediate score computations, preserving the scalability and strong empirical performance of BOSS in the static setting. We provide theoretical guarantees establishing the soundness of TS-BOSS under suitable assumptions, and we present an intermediate result that extends classical subgraph minimality results for permutation-based methods to the dynamic (time series) setting. Our experiments on synthetic data show that TS-BOSS is especially effective in high auto-correlation regimes, where it consistently achieves higher adjacency recall at comparable precision than standard constraint-based methods. Overall, TS-BOSS offers a high-performing, scalable approach for time series causal discovery and our results provide a principled bridge for extending sparsity-based, permutation-driven causal learning theory to dynamic settings.

Irene Gema Castillo Mansilla, Urmi Ninad

University of Potsdam, Potsdam, Germany · Department of Computer Science, University of Potsdam, Potsdam, Germany

Time-Series Causal Discovery Bayesian Networks Score-Based Causal Discovery Causal Discovery Causal Structure Learning

Mar 4, 2026

JANUS: Structured Bidirectional Generation for Guaranteed Constraints and Analytical Uncertainty

High-stakes synthetic data generation faces a fundamental Quadrilemma: achieving Fidelity to the original distribution, Control over complex logical constraints, Reliability in uncertainty estimation, and Efficiency in computational cost -- simultaneously. State-of-the-art Deep Generative Models (CTGAN, TabDDPM) excel at fidelity but rely on inefficient rejection sampling for continuous range constraints. Conversely, Structural Causal Models offer logical control but struggle with high-dimensional fidelity and complex noise inversion. We introduce JANUS (Joint Ancestral Network for Uncertainty and Synthesis), a framework that unifies these capabilities using a DAG of Bayesian Decision Trees. Our key innovation is Reverse-Topological Back-filling, an algorithm that propagates constraints backwards through the causal graph, achieving 100% constraint satisfaction on feasible constraint sets without rejection sampling. This is paired with an Analytical Uncertainty Decomposition derived from Dirichlet priors, enabling 128x faster uncertainty estimation than Monte Carlo methods. Across 15 datasets and 523 constrained scenarios, JANUS achieves state-of-the-art fidelity (Detection Score 0.497), eliminates mode collapse on imbalanced data, and provides exact handling of complex inter-column constraints (e.g., Salary_offered >= Salary_requested) where baselines fail entirely.

Taha Racicot

Universit´e Laval

Constrained Generative Modeling Synthetic Tabular Data Generation Bayesian Networks Uncertainty Decomposition

May 21, 2025

Extracting Probabilistic Knowledge from Large Language Models for Bayesian Network Parameterization

In this work, we evaluate the potential of Large Language Models (LLMs) in building Bayesian Networks (BNs) by approximating domain expert priors. LLMs have demonstrated potential as factual knowledge bases; however, their capability to generate probabilistic knowledge about real-world events remains understudied. We explore utilizing the probabilistic knowledge inherent in LLMs to derive probability estimates for statements regarding events and their relationships within a BN. Using LLMs in this context allows for the parameterization of BNs, enabling probabilistic modeling within specific domains. Our experiments on eighty publicly available Bayesian Networks, from healthcare to finance, demonstrate that querying LLMs about the conditional probabilities of events provides meaningful results when compared to baselines, including random and uniform distributions, as well as approaches based on next-token generation probabilities. We explore how these LLM-derived distributions can serve as expert priors to refine distributions extracted from data, especially when data is scarce. Overall, this work introduces a promising strategy for automatically constructing Bayesian Networks by combining probabilistic knowledge extracted from LLMs with real-world data. Additionally, we establish the first comprehensive baseline for assessing LLM performance in extracting probabilistic knowledge.

Aliakbar Nafar, Kristen Brent Venable, Zijun Cui +1

Michigan State University · Florida Institute for Human and Machine Cognition (IHMC) University of West Florida

Bayesian Networks Probabilistic Graphical Models Probability Elicitation from Language Models