Ultra-efficient MCMC for Bayesian longitudinal functional data analysis

<b>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. Supplementary materials for this article are available online.</b>

函数型混合模型（functional mixed models）广泛适用于带有相依函数型数据的回归分析，涵盖带有标量预测变量的纵向函数型数据场景。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精准近似，无法同时兼顾两者。我们提出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，我们的算法可联合采样固定效应回归函数，以及所有个体专属与重复专属的随机效应函数。尤为关键的是，该联合采样器在保留计算可扩展性的同时，优化了这些核心参数的采样效率。值得一提的是，我们的新型MCMC采样算法甚至超越了当前函数型混合模型领域最先进的频率估计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。仿真实验结果表明，在几乎所有仿真场景下，相较于其他同类竞争方法，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项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水平的关联关系。本文的补充材料可在线获取。