A Bayesian Nonparametric Regression Model With Normalized Weights - A Study of Hippocampal Atrophy in Alzheimer’s Disease

Hippocampal volume is one of the best established biomarkers for Alzheimer’s disease. However, for appropriate use in clinical trials research, the evolution of hippocampal volume needs to be well understood. Recent theoretical models propose a sigmoidal pattern for its evolution. To support this theory, the use of Bayesian nonparametric regression mixture models seems particularly suitable due to the flexibility that models of this type can achieve and the unsatisfactory predictive properties of semiparametric methods. In this article, our aim is to develop an interpretable Bayesian nonparametric regression model which allows inference with combinations of both continuous and discrete covariates, as required for a full analysis of the dataset. Simple arguments regarding the interpretation of Bayesian nonparametric regression mixtures lead naturally to regression weights based on normalized sums. Difficulty in working with the intractable normalizing constant is overcome thanks to recent advances in MCMC methods and the development of a novel auxiliary variable scheme. We apply the new model and MCMC method to study the dynamics of hippocampal volume, and our results provide statistical evidence in support of the theoretical hypothesis

Recentered importance sampling with applications to Bayesian model validation

Since its introduction in the early 1990s, the idea of using importance sampling (IS) with Markov chain Monte Carlo (MCMC) has found many applications. This article examines problems associated with its application to repeated evaluation of related posterior distributions with a particular focus on Bayesian model validation. We demonstrate that, in certain applications, the curse of dimensionality can be reduced by a simple modification of IS. In addition to providing new theoretical insight into the behavior of the IS approximation in a wide class of models, our result facilitates the implementation of computationally intensive Bayesian model checks. We illustrate the simplicity, computational savings, and potential inferential advantages of the proposed approach through two substantive case studies, notably computation of Bayesian p-values for linear regression models and simulation-based model checking. Supplementary materials including the Appendix and the R code for Section 3.1.2 are available online

Bayesian Nonparametric Analysis for a Generalized Dirichlet Process Prior

Bayesian nonparametric inference, Dirichlet process, generalized gamma convolutions, Lauricella hypergeometric functions, means of random probability measures, predictive distributions, 62F15, 60G57,

The local Dirichlet process

Dependent Dirichlet process, Blocked Gibbs sampler, Mixture model, Non-parametric Bayes, Stick-breaking representation,