While Bayesian functional mixed models have been shown effective to model functional data with various complex structures, their application to extremely high-dimensional data is limited due to computational challenges involved in posterior sampling. We introduce a new computational framework that enables ultra-fast approximate inference for high-dimensional data in functional form. This framework adopts parsimonious basis to represent functional observations, which facilitates efficient compression and parallel computing in basis space. Instead of performing expensive Markov chain Monte Carlo sampling, we approximate the posterior distribution using variational Bayes and adopt a fast iterative algorithm to estimate parameters of the approximate distribution. Our approach facilitates a fast multiple testing procedure in basis space, which can be used to identify significant local regions that reflect differences across groups of samples. We perform two simulation studies to assess the performance of approximate inference, and demonstrate applications of the proposed approach by using a proteomic mass spectrometry dataset and a brain imaging dataset. Supplementary materials are available online.
Keywords: Approximate Bayesian Inference; Distributed Inference; Functional Data Analysis; Parallel Computing; Variational Bayes.