Guoliang Charlie Cao
Senior Director of StatisticsTakeda Development Center Americas
Bayesian Nonparametric Mixture Modeling
This dissertation explores a Bayesian nonparametric approach to mixture modeling and the use of the Gibbs sampling scheme to approximate posterior estimates. The predictive distribution is modeled as a mixture of normal distributions by using a Dirichlet process prior for the unknown means and variances. The definition and some properties of mixtures of Dirichlet processes are reviewed. Analytically evaluating the predictive distribution is very tedious and difficult in this case. An approximation based on Monte Carlo integration is proposed. The Gibbs sampling method based on unknown means and variances is used and convergence based on the configuration space is proved. Different models are compared based on simulated data. Order statistics are used to solve the problem of identifiability. Recurrence relations are derived to calculate the distribution of the order statistics in the case of independent non-identically distributed random variables. The practicality of the nonparametric Bayesian analysis for mixture modeling is showed. Various methodological and computational aspects are developed. Mixture analysis for grouped data is discussed. Some comparisons between Bayesian methods and classical methods are described. A Bayesian analysis of mixtures of mixtures is introduced and illustrated in the context of neurological response analysis.