Ultra-efficient MCMC for Bayesian longitudinal functional data analysis

TL;DR

This paper presents a novel, highly efficient MCMC algorithm for Bayesian longitudinal functional data analysis that outperforms existing methods in speed and accuracy, enabling scalable and precise inference.

Contribution

Introduces a new MCMC sampling strategy with a novel blocking structure and orthogonalized basis, achieving both scalability and high-quality Bayesian inference for functional mixed models.

Findings

Surpasses state-of-the-art algorithms in speed and accuracy

Improves point estimation and interval coverage in simulations

Effectively analyzes large physical activity dataset

Abstract

Functional mixed models are widely useful for regression analysis with dependent functional data, including longitudinal functional data with scalar predictors. However, existing algorithms for Bayesian inference with these models only provide either scalable computing or accurate approximations to the posterior distribution, but not both. We introduce a new MCMC sampling strategy for highly efficient and fully Bayesian regression with longitudinal functional data. Using a novel blocking structure paired with an orthogonalized basis reparametrization, our algorithm jointly samples the fixed effects regression functions together with all subject- and replicate-specific random effects functions. Crucially, the joint sampler optimizes sampling efficiency for these key parameters while preserving computational scalability. Perhaps surprisingly, our new MCMC sampling algorithm even surpasses state-of-the-art algorithms for frequentist estimation and variational Bayes approximations for functional mixed models -- while also providing accurate posterior uncertainty quantification -- and is orders of magnitude faster than existing Gibbs samplers. Simulation studies show improved point estimation and interval coverage in nearly all simulation settings over competing approaches. We apply our method to a large physical activity dataset to study how various demographic and health factors associate with intraday activity.