Ultra-Efficient MCMC for Bayesian Longitudinal Functional Data Analysis

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 reparameterization, 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. Supplementary materials for this article are available online.

函数型混合模型（functional mixed models）在相依函数型数据的回归分析中应用广泛，这类数据涵盖带有标量预测变量的纵向函数型数据。然而，现有针对这类模型的贝叶斯推断算法，仅能实现可扩展计算或是后验分布的精准近似，无法同时兼顾两者。我们提出了一种全新的马尔可夫链蒙特卡洛（Markov Chain Monte Carlo, MCMC）采样策略，可实现纵向函数型数据下的高效全贝叶斯回归分析。通过结合新颖的分块结构与正交基重参数化方法，我们的算法可同时采样固定效应回归函数，以及所有个体及重复测量专属的随机效应函数。至关重要的是，该联合采样器在保障计算可扩展性的同时，还针对这些关键参数优化了采样效率。令人意外的是，我们提出的新型MCMC采样算法，甚至在性能上超越了现有用于函数型混合模型的顶尖频率学派估计算法与变分贝叶斯近似算法，同时还能提供精准的后验不确定性量化结果，且运行速度比现有吉布斯采样器快数个数量级。模拟研究结果表明，在几乎所有模拟场景中，相较于同类竞争方法，我们的方法在点估计与区间覆盖率上均有提升。我们将所提方法应用于一个大型体力活动数据集，以探究各类人口统计学与健康因素与日内活动模式之间的关联。本文的补充材料可在线获取。