An Empirical Bayes Algorithm for Variable Selection With Applications in Genetic Fine-Mapping

Identifying causal variants within genome-wide association study loci is challenging due to linkage disequilibrium, multi-signal architectures, and the need for calibrated uncertainty at a large scale. We formulate fine-mapping as Bayesian variable selection with binary inclusion indicators and propose a variational EM algorithm that learns feature-specific prior inclusion probabilities via empirical Bayes (EmpBVS). Our E-step delivers closed-form updates for the variational Gaussian factor on effects and the noise precision, while the M-step updates per-SNP inclusion priors by maximizing the lower bound of marginal data likelihood. Irrelevant SNPs are shrunk out automatically as their weights contract to zero, yielding threshold-free selection. We establish a computational sparsity result showing geometric decay of null inclusion weights across iterations, and statistical consistency, where posterior mass concentrates on the true model and variational means consistently recover effect sizes. Simulations and comparative experiments demonstrate accurate posterior inclusion probabilities and credible sets with competitive runtime. Our framework thus preserves automatic relevance learning while providing discrete selections and rigorous guarantees tailored to genetic fine-mapping.