Variational Bayesian inference for functional data clustering and survival data analysis

Variational Bayesian inference is a method to approximate the posterior distribution under a Bayesian model analytically. As an alternative to Markov Chain Monte Carlo (MCMC) methods, variational inference (VI) produces an analytical solution to an approximation of the posterior but have a lower computational cost compared to MCMC methods. The main challenge of applying VI comes from deriving the equations used to update the approximated posterior parameters iteratively, especially when dealing with complex data. In this thesis, we apply the VI to the context of functional data clustering and survival data analysis. The main objective is to develop novel VI algorithms and investigate their performance under these complex statistical models.\nIn functional data analysis, clustering aims to identify underlying groups of curves without prior group membership information. The first project in this thesis presents a novel variational Bayes (VB) algorithm for simultaneous clustering and smoothing of functional data using a B-spline regression mixture model with random intercepts. The deviance information criterion is employed to select the optimal number of clusters.\nThe second project shifts focus to survival data analysis, proposing a novel mean-field VB algorithm to infer parameters of the log-logistic accelerated failure time (AFT) model. To address intractable calculations, we propose and incorporate a piecewise approximation technique into the VB algorithm, achieving Bayesian conjugacy.\nThe third project is motivated by invasive mechanical ventilation data from intensive care units (ICUs) in Ontario, Canada, which form multiple clusters. We assume that patients within the same ICU cluster are correlated. Extending the second project's methodology, a shared frailty log-logistic AFT model is introduced to account for intra-cluster correlation through a cluster-specific random intercept. A novel and fast VB algorithm for model parameter inference is presented.\nExtensive simulation studies assess the performance of the proposed VB algorithms, comparing them with other methods, including MCMC algorithms. Applications to real data, such as ICU ventilation data from Ontario, illustrate the methodologies' practical use. The proposed VB algorithms demonstrate excellent performance in clustering functional data and analyzing survival data, while significantly reducing computational cost compared to MCMC methods.