# Empirical Bayes

Standard cryo-EM pipelines first reconstruct a 3D density from 2D projections and only then fit an atomic model, so atomic-coordinate priors never reach the most under-determined stage. EB-FM trains a flow-matching prior inside a Monte Carlo stochastic-approximation expectation-maximization loop, repeatedly inverting the nonlinear imaging forward model during training, so no clean structures are needed for pretraining. On simulated cryo-EM data with known poses it recovers conformational distributions directly in atomic-coordinate space for four proteins, though predicted motion amplitudes stay below the molecular-dynamics ground truth.

Standard cryo-EM pipelines first reconstruct a 3D density from 2D projections and only then fit an atomic model, so atomic-coordinate priors never reach the most under-determined stage. EB-FM trains a flow-matching prior inside a Monte Carlo stochastic-approximation expectation-maximization loop, repeatedly inverting the nonlinear imaging forward model during training, so no clean structures are needed for pretraining. On simulated cryo-EM data with known poses it recovers conformational distributions directly in atomic-coordinate space for four proteins, though predicted motion amplitudes stay below the molecular-dynamics ground truth.

The authors introduce Hierarchical Spectral Shrinkage (HSS), an empirical-Bayes framework that shrinks each task's leading spectral directions toward a data-adaptively learned common basis with shrinkage varying across tasks and spectral components, yielding the posterior mean of singular vectors in a surrogate Bayesian regression; across synthetic experiments and three immune-cell gene-expression datasets (sample sizes 156, 628, 146), HSS improves estimation accuracy and out-of-sample performance over separate, fully pooled, and shared-subspace approaches.

The authors introduce Hierarchical Spectral Shrinkage (HSS), an empirical-Bayes framework that shrinks each task's leading spectral directions toward a data-adaptively learned common basis with shrinkage varying across tasks and spectral components, yielding the posterior mean of singular vectors in a surrogate Bayesian regression; across synthetic experiments and three immune-cell gene-expression datasets (sample sizes 156, 628, 146), HSS improves estimation accuracy and out-of-sample performance over separate, fully pooled, and shared-subspace approaches.