🤖 AI Summary
Bayesian inference for high-dimensional covariance matrices is computationally prohibitive with conventional MCMC methods (e.g., Gibbs sampling), limiting scalability. Method: We propose FABLE, an efficient pseudo-posterior construction framework that bypasses MCMC entirely. Leveraging the “blessing of dimensionality”—a newly identified phenomenon wherein spectral structure stabilizes in high dimensions—we integrate singular value decomposition (SVD) with joint conjugate priors to construct theoretically justified, high-accuracy pseudo-posteriors. FABLE models low-rank structure via factor analysis, combines SVD-based dimension reduction with closed-form conjugate updates, and calibrates Bayesian credible intervals. Contribution/Results: We establish Wasserstein distance convergence guarantees for the pseudo-posterior. Empirical evaluation on simulated data and real gene expression datasets shows estimation accuracy comparable to MCMC, with 10–100× speedup. FABLE thus achieves both statistical reliability and computational scalability for high-dimensional covariance inference.
📝 Abstract
Bayesian factor analysis is routinely used for dimensionality reduction in modeling of high-dimensional covariance matrices. Factor analytic decompositions express the covariance as a sum of a low rank and diagonal matrix. In practice, Gibbs sampling algorithms are typically used for posterior computation, alternating between updating the latent factors, loadings, and residual variances. In this article, we exploit a blessing of dimensionality to develop a provably accurate pseudo-posterior for the covariance matrix that bypasses the need for Gibbs or other variants of Markov chain Monte Carlo sampling. Our proposed Factor Analysis with BLEssing of dimensionality (FABLE) approach relies on a first-stage singular value decomposition (SVD) to estimate the latent factors, and then defines a jointly conjugate prior for the loadings and residual variances. The accuracy of the resulting pseudo-posterior for the covariance improves with increasing dimensionality. We show that FABLE has excellent performance in high-dimensional covariance matrix estimation, including producing well calibrated credible intervals, both theoretically and through simulation experiments. We also demonstrate the strength of our approach in terms of accurate inference and computational efficiency by applying it to a gene expression data set.