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用途。然而，现有针对此类模型的贝叶斯推断算法要么只能实现可扩展计算，要么仅能提供后验分布的精确近似，而无法同时满足两者。我们提出一种新的MCMC采样策略，用于实现纵向函数数据的高效全贝叶斯回归。该算法采用新颖的块状结构配合正交化基函数重参数化，可联合采样固定效应回归函数以及所有受试者特异性和重复测量特异性随机效应函数。关键在于，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。令人意外的是，我们提出的新MCMC采样算法甚至超越了用于函数混合模型的频率学派最先进估计方法和变分贝叶斯近似方法——同时还能提供精确的后验不确定性量化——且速度比现有吉布斯采样器快数个数量级。模拟研究表明，在几乎所有模拟场景下，该方法相比竞争方法均能改进点估计精度和区间覆盖率。我们将该方法应用于大规模身体活动数据集，以探究人口统计学和健康因素如何与日内活动模式相关联。