Efficient and Accurate Variational Inference for Multilevel Threshold Autoregressive Models in Intensive Longitudinal Data

Recent technological advancements have facilitated the collection of intensive longitudinal data (ILD), comprising repeated measurements from the same individual. The threshold autoregressive (TAR) model is commonly utilized to capture the dynamic outcome process in ILD, with autoregressive parameters varying based on outcome variable levels. For ILD data from multiple individuals, multilevel TAR models have been proposed, with a Bayesian approach recommended for parameter estimation. However, fitting multilevel TAR models can present computational challenges. In this study, we introduce a mean field variational Bayes (MFVB) algorithm as an alternative to traditional Bayesian inference. By optimizing to approximate posterior densities, variational Bayes aims to find the best approximation within a defined set of distributions rather than asymptotically converging to the true posterior. Simulation results demonstrate that our MFVB algorithm is significantly faster than the standard Markov Chain Monte Carlo (MCMC) approach. Additionally, the simulation results reveal that increasing the number of individuals or the number of time points improves the accuracy of parameter estimates using the MFVB method, suggesting that sufficient data are critical for accurate parameter estimation, especially for complex models like multilevel threshold autoregressive models. When the algorithm was applied to real-world data, it significantly outperformed MCMC in speed and accuracy. In summary, the MFVB algorithm offers a notably faster and consistent alternative to MCMC for intricate and large-scale inference in intensive longitudinal data models, underscoring its potential as a swift and efficient tool for data analysis.