Disentangling Latent Risk Pathways
via Bayesian Hypergraph Inference

Official implementation · Yale University

🌐 Project page  ·  📄 Paper (arXiv)  ·  📌 BibTeX

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🧩 Overview

BHPI reframes multi-disease modeling as inferring a latent hypergraph: diseases group into overlapping pathways (hyperedges), and each risk factor acts on pathways rather than individual diseases. A disease's per–risk-factor effect is composed from the pathways it belongs to:

$$\beta_{j,v} = d_v^{-1} \sum_{e} H_{v,e} \mu_{j,e}$$

A repulsion prior keeps the discovered pathways parsimonious and identifiable, and a structured variational inference scheme (Pólya–Gamma augmentation + CAVI) preserves the existence → membership → effect logic for calibrated posterior uncertainty over both the disease groupings and the risk-factor effects.

📁 Repository structure

BHPI/
├── BHPI.m              # core algorithm: repulsion-aware coordinate-ascent VI
├── simulate_design.m   # entry point: synthetic experiments + structural recovery
├── helper/             # synthetic data generation, hypergraph init, repulsion utilities
└── docs/               # project page (served via GitHub Pages)

🚀 Getting started

Requirements: MATLAB (R2023a+) with the Statistics and Machine Learning Toolbox.

Reproduce the synthetic structure-recovery experiments:

simulate_design

This simulates data from a known latent hypergraph, fits BHPI, and reports structural recovery (incidence H and effect γ/μ) alongside predictive AUC against the baselines.

🛠 Usage

Initialize the variational parameters, fit the model, then predict and evaluate — as in simulate_design.m:

% 1. Initialize variational parameters (NNMF initialization recommended)
[initials] = cavi_initialization(seed_init, initial_method, E_hat, X_train, Y_train, []);

% 2. Fit the BHPI model
model = BHPI(X_train, Y_train, E_hat, max_iter, ...
             seed_init, initials, omega_repulsion, staged, ...
             fix_z, z_constraint, sigma2_alpha, ...
             warmup_iters, batch_size, t0, weights, tol, verbose);

% 3. Predict on held-out data
eta_val  = X_val * model.beta + model.alpha_mean;
prob_val = 1 ./ (1 + exp(-eta_val));

% 4. Score per-disease AUROC
AUROC = NaN(1, V);
for v = 1:V
    [~, ~, ~, AUROC(v)] = perfcurve(Y_val(:, v), prob_val(:, v), 1);
end
mean_auroc = mean(AUROC);

Key parameters

Argument	Meaning
E_hat	Upper bound on the number of latent hyperedges; the model self-regularizes to fewer.
omega_repulsion	Repulsion strength; > 0 disentangles redundant pathways.
initials	Starting values for the variational parameters (from cavi_initialization).
model.beta	Learned disease-specific risk-factor effects.

See the header of BHPI.m for the full argument list (staged, fix_z, warmup_iters, batch_size, …).

⏱️ Runtime & complexity

Phase	Per-iteration complexity	UK Biobank ($N \approx 277\text{K}$)
Training	$\mathcal{O}(N \cdot E \cdot (P + V))$	~74 min · ~28 GB peak
Inference	efficient matrix ops	$< 0.1$ ms / sample

Measured on 4 × Intel Xeon 6342 cores, 60 GB RAM; inference latency is on par with logistic regression.

🗄️ Data availability

The synthetic experiments are fully reproducible from this repository; the paper's real-data results use the UK Biobank, which requires approved access and cannot be redistributed here.

✒️ Citation

@inproceedings{ding2026bhpi,
  title     = {Disentangling Latent Risk Pathways via Bayesian Hypergraph Inference},
  author    = {Ding, Shengxian and Gao, Haonan and Liu, Pangpang and Tian, Xinyuan and Zhao, Yize},
  booktitle = {Proceedings of the 43rd International Conference on Machine Learning (ICML)},
  series    = {Proceedings of Machine Learning Research},
  publisher = {PMLR},
  year      = {2026},
  eprint    = {2606.07677},
  archivePrefix = {arXiv}
}

📜 License

Released under the MIT License.