NExON-Bayes: Bayesian Network Estimation

• NExON-Bayes is a Bayesian framework that extends graphical spike-and-slab models to jointly incorporate ordinal covariate effects for accurate network estimation in high-dimensional omics data.
• It employs a deterministic variational Bayes algorithm, enabling scalable inference and uncertainty quantification across heterogeneous network structures.
• The method demonstrated superior edge recovery and biological interpretability in simulations and real breast carcinoma proteomic data, facilitating biomarker discovery.

NExON-Bayes is a Bayesian methodological framework for estimating sample-specific conditional dependence structures in high-dimensional omic networks, particularly in scenarios characterized by heterogeneous populations and ordinal covariates (such as disease stage). The model extends the graphical spike-and-slab paradigm, enabling the joint incorporation of ordinal covariate information to both inform network structure and explicitly quantify covariate effects on network connectivity. This approach overcomes limitations of conventional graphical models that assume homogeneity, providing both improved accuracy and interpretability in the presence of disease heterogeneity. To ensure scalability in high-dimensional settings, NExON-Bayes employs a deterministic variational inference algorithm, facilitating efficient model fitting and uncertainty quantification for large omics datasets (Feest et al., 29 Aug 2025).

1. Model Framework: Joint GGM with Ordinal Covariates

NExON-Bayes builds on the Gaussian graphical model (GGM) representation. For $P$ observed omic variables and $A$ ordinal levels (indexed by $a$), each group-specific sample $y_n^{(a)}$ is modeled as

$$y_n^{(a)} \sim \mathcal{N}_P(0, (\Omega^{(a)})^{-1}),$$

where $\Omega^{(a)}$ is the precision matrix at level $a$ of the ordinal covariate.

To induce sparsity and enable model selection over the edge set $\mathcal{E}$, a graphical spike-and-slab prior is imposed on each off-diagonal precision entry: $$\Omega_{ij}^{(a)}|\delta_{ij}^{(a)} \sim \delta_{ij}^{(a)} \mathcal{N}(0, \nu_1^2) + (1 - \delta_{ij}^{(a)}) \mathcal{N}(0, \nu_0^2),$$ with $\nu_0^2 \ll \nu_1^2$ to drive non-relevant edges toward zero.

Crucially, NExON-Bayes introduces a probit regression submodel for the edge-inclusion indicators to capture ordinal covariate effects: $$\delta_{ij}^{(a)} | \zeta_{ij}, \beta_{ij} \sim \mathrm{Bernoulli}\left\{\Phi(\zeta_{ij} + a \beta_{ij})\right\},$$ where $\Phi$ denotes the standard normal CDF, $\zeta_{ij}$ is a network-edge-specific intercept, and $\beta_{ij}$ is the coefficient encoding the ordinal covariate dependency. The sign and magnitude of $\beta_{ij}$ quantify how the probability of edge inclusion responds to the covariate, such that a positive (negative) $\beta_{ij}$ means that the $(i,j)$ edge is increasingly likely (unlikely) to appear in higher ordinal strata.

2. Inference Methodology: Variational Bayes Expectation Conditional Maximization

To ensure computational tractability in high-dimensional settings (e.g., $P \gg 100$), NExON-Bayes utilizes a variational Bayes expectation conditional maximization (VBECM) algorithm. This approach factorizes the joint posterior over parameters into tractable components: $$q(\Omega, \Theta) = \prod_{a \in \mathcal{A}} q(\Omega^{(a)}) \prod_{a \in \mathcal{A}} \prod_{i<j} q(\delta_{ij}^{(a)}, z_{ij}^{(a)}) \prod_{i<j} q(\zeta_{ij}) q(\sigma^{-2}) \prod_{i<j} q(\beta_{ij}),$$ where $z_{ij}^{(a)}$ is an introduced latent variable such that $\delta_{ij}^{(a)} = I\{ z_{ij}^{(a)} > 0 \}$ and

$$z_{ij}^{(a)} | \zeta_{ij}, \beta_{ij} \sim \mathcal{N}(\zeta_{ij} + a\beta_{ij}, 1).$$

The variational lower bound (ELBO) $\mathcal{L}(q)$ is maximized through block-wise updates on each variational factor. The update of the blockwise $\Omega^{(a)}$ is cast as a conditional maximization, leveraging the structure of the spike-and-slab prior across groups and the shared ordinal components. This approach substantially accelerates convergence compared to MCMC and enables scalability to large omics networks.

3. Performance Evaluation and Comparative Analysis

In simulation experiments simulating $A=4$ networks with $P = 100$ nodes and $N = 150$ samples per network, NExON-Bayes demonstrated substantially higher area under the curve (AUC) for edge recovery than conventional single-network spike-and-slab (SSL) models. For instance, Network 2 showed an AUC improvement from 0.870 (SSL) to 0.967 (NExON-Bayes). In addition, when compared to other state-of-the-art methods that use covariate information (such as the Bayesian Joint Spike-and-Slab Graphical Lasso and covariate-dependent graphical estimation), NExON-Bayes exhibited higher recall and comparable or superior precision, particularly in regimes with weak signal or low sample size per stratum.

Recovery of the $\beta_{ij}$ coefficient matrix was accurate, with strong correspondence between estimated and true coefficients. This capability enables detection of edges whose inclusion probabilities shift monotonically with the ordinal covariate. Borrowing strength via the $\beta_{ij}$ coefficients enhances sensitivity to subtle edge changes across stratified networks.

4. Application to Breast Carcinoma Omics Data

NExON-Bayes was applied to proteomic data from The Cancer Genome Atlas (TCGA), consisting of reverse phase protein array measurements for 131 proteins across breast carcinoma patients with known tumor stage (I/II/III). Key findings include:

• The structure of proteomic networks changes markedly with disease progression, with evidence of both emerging and vanishing conditional dependencies between proteins as stage increases.
• The distribution of $\beta_{ij}$ coefficients revealed protein subnetworks whose connectivity becomes more or less probable as tumor stage increases. Subnetworks with large positive $\beta_{ij}$ were enriched in biological pathways related to apoptosis, TNF$\alpha$ signaling, and ATM signaling—implicating these processes in the molecular shift accompanying breast cancer progression.
• Cross-validation of detected edges against the STRING database for protein–protein interactions confirmed the plausibility of many findings, thus supporting the method’s biological interpretability and statistical reliability.

5. Ordinal Covariate Modeling and Biological Interpretation

By explicitly parametrizing $$\delta_{ij}^{(a)} \sim \mathrm{Bernoulli}\{\Phi(\zeta_{ij} + a \beta_{ij})\}$$ with edge-specific $\beta_{ij}$, NExON-Bayes permits direct inference and visualization of network dynamics with respect to ordinal sample features. A positive $\beta_{ij}$ indicates monotonic activation of the edge along the covariate axis (e.g., increasing prevalence with advancing disease), while negative values identify edges whose likelihood diminishes. This framework enables researchers to isolate subnetworks modulated by clinical or experimental factors, thereby deepening understanding of underlying molecular processes and aiding biomarker or pathway discovery.

6. Practical Implementation and Usage

NExON-Bayes is released as a user-friendly R package, available at github.com/jf687/NExON. The package supports preprocessing of omics data with ordinal covariate annotation, execution of the VBECM inference algorithm, and convenient visualization of the inferred networks and $\beta_{ij}$ matrices. Researchers can tune sparsity through regularization parameters, use provided tutorials, and apply gene set enrichment analyses to subnetworks defined by covariate modulation.

The package empowers applied researchers to efficiently estimate and interpret covariate-modulated networks in high-dimensional omic studies, with particular relevance to disease heterogeneity and progression analysis.

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In summary, NExON-Bayes advances Bayesian network estimation by extending graphical spike-and-slab models to explicitly account for ordinal covariate effects using a joint GGM/probit regression submodel, scalable variational inference, and statistically principled edge selection and effect quantification. The framework has demonstrated improved performance on simulations and on real cancer proteomics, and provides actionable insights into the molecular evolution of disease across covariate-defined strata (Feest et al., 29 Aug 2025).