Fast approximate Bayesian estimation algorithm for discrete and continuous time models in intensive longitudinal data

Abstract

Intensive longitudinal data (ILD) consist of frequent repeated measurements collected over time in natural settings. These data often include irregular observation times and complex temporal dependence, which require flexible statistical methods for proper analysis. Continuous-time (CT) modeling frameworks are particularly useful in this context because they allow changes in the underlying process to occur at any time rather than at fixed intervals. The continuous-time mixed hidden Markov model (CT-MixHMM) provides a flexible approach for modeling latent disease states and their transitions while accounting for individual-level heterogeneity. Traditional Bayesian estimation methods, particularly Markov Chain Monte Carlo (MCMC), however, are often computationally demanding when applied to CT-MixHMM models and large ILD datasets. This study develops a computationally efficient approximate Bayesian estimation framework for CT models applied to ILD. Specifically, a Sequential Variational Bayes (SVB) algorithm was proposed for the multivariate version of CT-MixHMM model. To improve convergence and numerical stability, a damped Recursive variational Gaussian approximation algorithm for dependent mixture (R-VGADM) model was introduced under SVB framework. The proposed algorithm was evaluated through simulation studies and applied to real data from the Manitoba Follow-Up Study (MFUS). Model performance was assessed using prediction accuracy, hidden state classification, mean absolute percentage error, posterior predictive checks, and computational efficiency. Results were compared with those obtained using MCMC estimation. Both simulation and real data analyses indicated that the SVB algorithm accurately captures partial absorbing state’s transition probability, and achieves faster computation and stable convergence, particularly for large datasets. In the MFUS application, age was associated with progression across hypertension stages, increasing treatment needs and mortality risk. Overall, the proposed SVB approach provides an efficient alternative to traditional Bayesian methods for analyzing complex intensive longitudinal health data.