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.

翻译：函数混合模型在包含标量预测变量的纵向函数数据等依赖函数数据的回归分析中具有广泛应用。然而，现有用于这些模型的贝叶斯推断算法要么仅能提供可扩展计算，要么仅能实现后验分布的精确近似，但无法同时兼顾两者。我们提出了一种新的MCMC采样策略，用于对纵向函数数据进行高效且全贝叶斯回归分析。通过采用新颖的分块结构并搭配正交化基函数重参数化，我们的算法能联合采样固定效应回归函数以及所有受试者和重复测量特定的随机效应函数。关键之处在于，这种联合采样器在保持计算可扩展性的同时，优化了这些关键参数的采样效率。令人瞩目的是，我们的新MCMC采样算法甚至超越了用于函数混合模型的频率学派估计和变分贝叶斯近似的先进算法——同时还能提供精确的后验不确定性量化——并且比现有吉布斯采样器快数个数量级。模拟研究表明，在几乎所有模拟设置下，该方法在点估计和区间覆盖率方面均优于竞争方法。我们将该方法应用于一个大型身体活动数据集，以研究各种人口统计学和健康因素如何与日内活动水平相关联。