Approximate Bayesian Inference for Functional Mixed Effects Regression with Partially Observed Functional Covariates

Functional covariates are increasingly collected over time, such as in environmental monitoring studies. In practice, these covariates are often partially observed due to irregular measurement schedules, sensor failures, or temporal aggregation. Although functional mixed-effects regression has been extensively studied for longitudinal data, existing Bayesian approaches, such as variational Bayes (VB) and Markov chain Monte Carlo (MCMC), assume fully observed functional covariates, thereby overlooking the pervasive issue of functional data incompleteness. Recent methodological developments have addressed partially observed functional data under both continuous and randomly scattered missingness, primarily through reconstruction or descriptive tools such as functional depth measures. However, these approaches are not embedded within functional mixed-effects regression models and therefore do not propagate reconstruction uncertainty into inferential procedures. To bridge this gap, we develop a unified modeling framework for a functional mixed-effects regression model with partially observed functional covariates that jointly reconstruct incomplete functional trajectories and perform inference. This integration enables simultaneous evaluation of imputation quality, functional coefficient estimation, and model-based inference. Through simulation studies across varying numbers of subjects and repeated measurements, we compare VB and MCMC in terms of estimation accuracy and computational efficiency. The results show that the reconstructed functional covariates preserve key features of the underlying trajectories and that VB produces estimates comparable to MCMC while achieving substantial reductions in computation time. We further illustrate the proposed method using real environmental data, modeling weekly water levels with daily temperature curves as functional covariates and evaporation levels as scalar covariates, demonstrating its practical and computational efficiency.