Bayesian Networks in Systems Biology: Probabilistic Graph Models for Veterinary and Biological Inference

Introduction

Systems biology seeks to understand biological phenomena as emergent properties of interacting molecular, cellular, and organismal components. Probabilistic graphical models, particularly Bayesian networks (BNs), provide a rigorous mathematical framework for representing conditional dependencies among variables, learning network structures from data, and inferring causal relationships. In veterinary medicine, BNs have been applied to model host-pathogen interactions, antimicrobial resistance patterns, gene regulatory circuits, and multi-omics data integration across species. This article provides an exhaustive technical review of Bayesian networks in systems biology, with emphasis on methodological advances, computational challenges, and applications relevant to veterinary science and diagnostics.

Core Concepts of Bayesian Networks

A Bayesian network is a directed acyclic graph (DAG) whose nodes represent random variables and whose edges encode conditional probability dependencies. The joint probability distribution over all variables factorizes as the product of local conditional probability distributions given the parent nodes in the graph. For discrete variables, these distributions are typically stored as conditional probability tables (CPTs). Continuous variables often assume Gaussian distributions with linear dependencies.

The structure of the DAG imposes a set of conditional independence assumptions that can be tested against observed data. In systems biology, nodes may correspond to gene expression levels, protein abundances, metabolite concentrations, phenotypic traits, or pathogen presence indicators. Edges represent putative regulatory, physical, or statistical associations. Because the graph is acyclic, feedback loops cannot be directly modeled, although dynamic Bayesian networks (DBNs) extend the framework to handle temporal dependencies.

The core inferential tasks in BN applications are: (1) structure learning (identifying the DAG that best explains the data), (2) parameter estimation (learning the conditional probability distributions given a fixed structure), and (3) probabilistic inference (computing posterior distributions of unobserved variables given evidence). These tasks draw on both frequentist and Bayesian statistical principles.

Structure Learning and Inference Methods

Structure learning for BNs remains a computationally challenging NP-hard problem. Algorithms fall into three broad categories: constraint-based (using conditional independence tests), score-based (optimizing a scoring metric), and hybrid approaches. The Bayesian Dirichlet equivalent (BDe) score and the Bayesian Information Criterion (BIC) are common scoring functions. In systems biology, the number of variables often exceeds the number of samples, necessitating regularization or informative priors.

Recent methodological contributions address this challenge. Martin et al. [1] introduced a flexible prior framework on edge states (presence, absence, or undirected orientation) that improves the accuracy of approximate Bayesian inference for DAGs, particularly when sample sizes are limited. Gogoshin and Rodin [2] proposed a model uncertainty criterion that quantifies the reliability of learned edges by assessing the stability of structures across bootstrap resamples; they demonstrated that operating characteristics of this criterion are superior to single-model selection approaches. The same authors [3] later developed a minimum uncertainty principle for BN model selection, showing that selecting the model with the smallest expected posterior variance reduces overfitting and improves predictive performance in genomic datasets.

Active learning strategies can reduce the computational burden of structure learning. Sándor and Antal [4] described a Bayesian active learning framework that iteratively selects the most informative experiments (e.g., gene knockout or perturbation assays) to disambiguate candidate network structures, thereby accelerating the recovery of gene regulatory networks with fewer interventions.

Gene Regulatory Network Recovery from Multi-Omics Data

One of the most prominent applications of BNs in systems biology is the inference of gene regulatory networks (GRNs) from high-throughput transcriptomic, proteomic, and epigenomic data. The goal is to infer directed edges from transcription factors (TFs) to their target genes, as well as interactions among TFs.

Gupta et al. [5] developed a Bayesian inference framework for GRNs that assumes the system has reached a stochastic steady state. Their method models gene expression as drawn from the stationary distribution of a stochastic dynamical system and uses reversible jump Markov chain Monte Carlo (MCMC) to sample over network structures and kinetic parameters. This approach avoids the need for time-course data and is applicable to cross-sectional single-cell RNA-seq datasets.

Zhang et al. [6] introduced PRISM-GRN, a method that recovers GRNs from single-cell multi-omics data (e.g., scRNA-seq and scATAC-seq simultaneously). PRISM-GRN employs a Bayesian network structure that integrates chromatin accessibility and gene expression, using a probabilistic model to infer regulatory relationships while accounting for technical dropout and batch effects. The method outperformed alternative approaches on benchmark mammalian datasets and identified novel regulatory modules in hematopoietic differentiation.

Multi-omics data are often incomplete due to missing measurements across different omic layers. Howey et al. [7] addressed this problem by applying Bayesian network imputation methods to a diabetes cohort dataset. Their approach imputes missing values in a principled probabilistic manner while simultaneously learning network structure, enabling the identification of putative causal relationships among transcriptomic, metabolomic, and phenotypic variables.

Causal Discovery and Deep Learning Challenges

The distinction between correlation and causation is central to systems biology. Yeo and Selvarajoo [8] critically assessed the readiness of deep learning methods for causal discovery in biological systems. They argued that while deep learning can capture complex nonlinear relationships, it often lacks the interpretability and theoretical guarantees of Bayesian approaches. Combining deep learning with BN structure priors (e.g., using neural networks to parameterize conditional distributions) remains an active area of research, but challenges include the need for large sample sizes and the risk of learning spurious correlations in high-dimensional settings.

Antimicrobial Resistance Modeling in Veterinary Populations

Bayesian networks are increasingly used to model the spread and evolution of antimicrobial resistance (AMR) in animal populations. Rupasinghe et al. [9] applied BN models to assess AMR patterns of Streptococcus suis isolated from swine production systems in the United States. Their model incorporated variables such as production stage, geographic region, antimicrobial use history, and resistance phenotypes to multiple drug classes. The BN framework allowed identification of conditional dependencies among resistances (e.g., co-resistance between tetracyclines and macrolides) and estimation of the probability of multidrug resistance under different management scenarios. Such models directly inform Antimicrobial Resistance in Livestock-Associated Staphylococcus aureus surveillance and intervention strategies.

Critical Transitions and Multimodel Inference

Complex biological systems can undergo abrupt shifts when approaching a tipping point, such as disease onset or population collapse. Tong et al. [10] developed BCTI (Bayesian Critical Transition Identifier), a BN-based method that detects early warning signals of critical transitions. BCTI constructs a network of observable variables and monitors changes in conditional dependencies that precede a state shift. The method was validated on simulated gene regulatory networks and on experimental data from bacterial stress responses, demonstrating its potential for early detection of disease outbreaks in livestock.

Model uncertainty is a pervasive issue in systems biology. Linden-Santangeli et al. [11] proposed a Bayesian multimodel inference framework that averages predictions across multiple plausible network models, weighted by their posterior probabilities. This approach increased the certainty of model predictions compared to selecting a single best model, particularly when data are sparse or when multiple network structures explain the data nearly equally well.

Workflow for Bayesian Network Analysis in Systems Biology

The typical workflow for applying BNs to a systems biology problem can be summarized in the following diagram.

graph TD
    A[Multi-Omics Data Acquisition], > B[Data Preprocessing & Normalization]
    B, > C{Structure Learning}
    C, > D[Constraint-Based]
    C, > E[Score-Based]
    C, > F[Hybrid/Active Learning]
    D & E & F, > G[Model Selection & Validation]
    G, > H[Parameter Estimation]
    H, > I[Probabilistic Inference]
    I, > J[Biological Interpretation]
    J, > K[Experimental Validation]

The process begins with data collection, often from high-throughput platforms (RNA-seq, mass spectrometry, microarrays). Preprocessing includes normalization, missing value imputation (potentially using BN imputation methods [7]), and discretization if discrete BNs are used. Structure learning is performed using one of the algorithmic families, with careful consideration of sample size and model uncertainty [2, 3]. The selected model is then parameterized, and inference is conducted to answer specific biological questions, such as predicting the effect of a gene knockout or estimating the probability of resistance given a treatment history.

Applications in Veterinary Diagnostics and Infectious Disease

Bayesian networks are particularly suited to veterinary diagnostics because they naturally handle missing data, integrate heterogeneous information (clinical signs, laboratory results, pathogen genomic data), and provide probabilistic predictions that can guide decision-making. For example, a BN could be constructed to diagnose respiratory disease complex in poultry by linking causal factors (pathogen presence, environmental stress, immune status) to observed symptoms. Such a model would be informed by data from diseases like Avian Cholera in Waterfowl or Mycoplasma bovis in Feedlot Cattle. In aquaculture, BNs can model the interaction between host genetics, environmental parameters, and pathogen load, as exemplified in studies of Streptococcosis in Farmed Tilapia.

The integration of BN structure learning with active experiment design [4] is particularly promising for veterinary virology, where the number of possible pathogen-host interactions is vast but intervention experiments are expensive and ethically constrained. By suggesting the most informative perturbations (e.g., specific receptor blockades or cytokine treatments), active learning can maximize knowledge gain per experiment.

Comparison of Selected BN Methods in Systems Biology

The table below summarizes key methodological contributions from the cited literature, focusing on their data requirements and inferential goals.

This table highlights the diversity of BN approaches: some focus on structure learning under uncertainty [1, 2, 3], others on integrating disparate data types [6, 7], and still others on domain-specific applications such as AMR [9] or critical transitions [10].

Conclusion

Bayesian networks provide a mature yet evolving framework for systems biology that is well suited to the complexities of veterinary science. The ability to represent probabilistic dependencies, handle missing data, and quantify uncertainty makes BNs particularly valuable for modeling host-pathogen systems, antimicrobial resistance dynamics, and multi-omics data integration. Recent advances in structure learning (flexible priors, active learning, model uncertainty criteria), steady-state inference, and multi-omics integration have expanded the scope of BNs in biology. However, challenges remain, including scalability to thousands of variables, guaranteeing causal interpretability, and integrating deep learning components without sacrificing transparency. As demonstrated by the studies reviewed here, BNs are poised to become a standard tool in veterinary bioinformatics and systems-level diagnostics, enabling more precise risk assessment and intervention design.

References

[1] Martin EA, Patchigolla V, Fu AQ. Approximate Bayesian inference of directed acyclic graphs in biology with flexible priors on edge states. PLoS Comput Biol. 2026. URL: https://pubmed.ncbi.nlm.nih.gov/41838802/

[2] Gogoshin G, Rodin AS. Reliable Bayesian Network Structure Learning in Biomedical Applications: Model Uncertainty Criterion and Its Operating Characteristics. bioRxiv. 2025. URL: https://pubmed.ncbi.nlm.nih.gov/41280097/

[3] Gogoshin G, Rodin AS. Minimum uncertainty as Bayesian network model selection principle. BMC Bioinformatics. 2025. URL: https://pubmed.ncbi.nlm.nih.gov/40200184/

[4] Sándor D, Antal P. Efficient structure learning of gene regulatory networks with Bayesian active learning. BMC Bioinformatics. 2025. URL: https://pubmed.ncbi.nlm.nih.gov/40461994/

[5] Gupta A, Yoon R, Josić K. Bayesian Inference of Gene Regulatory Networks at Stochastic Steady State. bioRxiv. 2026. URL: https://pubmed.ncbi.nlm.nih.gov/41659677/

[6] Zhang W, Cao L, Gu X, et al. Recovering gene regulatory networks in single-cell multi-omics data with PRISM-GRN. Genome Res. 2026. URL: https://pubmed.ncbi.nlm.nih.gov/41067887/

[7] Howey R, Adam J, Adamski J, et al. Bayesian network imputation methods applied to multi-omics data identify putative causal relationships in a type 2 diabetes dataset containing incomplete data: An IMI DIRECT Study. PLoS Genet. 2025. URL: https://pubmed.ncbi.nlm.nih.gov/40663565/

[8] Yeo HC, Selvarajoo K. Are we ready for causal discovery in biological systems using deep learning? Brief Bioinform. 2026. URL: https://pubmed.ncbi.nlm.nih.gov/41911151/

[9] Rupasinghe R, Morgan Bustamante BL, Robbins RC, et al. Bayesian network models to assess antimicrobial resistance patterns of Streptococcus suis isolated from swine production systems in the United States between 2014-2021. PLoS Comput Biol. 2026. URL: https://pubmed.ncbi.nlm.nih.gov/41886440/

[10] Tong Y, Hong R, Yang N, et al. BCTI: a Bayesian network-based method for revealing critical transitions in complex biological systems. PeerJ. 2026. URL: https://pubmed.ncbi.nlm.nih.gov/41704237/

[11] Linden-Santangeli N, Zhang J, Kramer B, et al. Increasing certainty in systems biology models using Bayesian multimodel inference. Nat Commun. 2025. URL: https://pubmed.ncbi.nlm.nih.gov/40790297/