... In this paper we present two general types of Gibbs samplers that can be used to fit posteriors of Bayesian hierarchical models based on stick-breaking priors. The first type of Gibbs sampler, referred to as a Polya urn Gibbs sampler, is a generalized version of a widely used Gibbs sampling method currently employed for Dirichlet process computing. This method applies to stick-breaking priors with a known P'olya urn characterization; that is priors with an explicit and simple prediction rule. Our second method, the blocked Gibbs sampler, is based on a entirely different approach that works by directly sampling values from the posterior of the random measure. The blocked Gibbs sampler can be viewed as a more general approach as it works without requiring an explicit prediction rule. We find that the blocked Gibbs avoids some of the limitations seen with the Polya urn approach and should be simpler for non-experts to use.

This article considers Bayesian methods for density regression, allowing a random probability distribution to change flexibly with multiple predictors. The conditional response dis-tribution is expressed as a nonparametric mixture of parametric densities, with the mixture distri-bution changing according to location in the predictor space. A new class of priors for dependent random measures is proposed for the collection of random mixing measures at each location. The conditional prior for the random measure at a given location is expressed as a mixture of a Dirichlet process (DP) distributed innovation measure and neighboring random measures. This specifica-tion results in a coherent prior for the joint measure, with the marginal random measure at each location being a finite mixture of DP basis measures. Integrating out the infinite-dimensional col-lection of mixing measures, we obtain a simple expression for the conditional distribution of the subject-specific random variables, which generalizes the Pólya urn scheme. Properties are consid-ered and a simple Gibbs sampling algorithm is developed for posterior computation. The methods are illustrated using simulated data examples and epidemiologic studies.

We review the current state of nonparametric Bayesian inference. The discussion follows a list of important statistical inference problems, including density estimation, regression, survival analysis, hierarchical models and model validation. For each inference problem we review relevant nonparametric Bayesian models and approaches including Dirichlet process (DP) models and variations, Polya trees, wavelet based models, neural network models, spline regression, CART, dependent DP models, and model validation with DP and Polya tree extensions of parametric models. 1

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The Dirichlet process prior allows flexible nonparametric mixture modeling. The number of mixture components is not specified in advance and can grow as new data come in. However, the behavior of the model is sensitive to the choice of the parameters, including an infinite-dimensional distributional parameter G0 . Most previous applications have either fixed G0 as a member of a parametric family or treated G0 in a Bayesian fashion, using parametric prior specifications. In contrast, we have developed an adaptive nonparametric method for constructing smooth estimates of G0 . We combine this method with a technique for estimating #, the other Dirichlet process parameter, that is inspired by an existing characterization of its maximum-likelihood estimator. Together, these estimation procedures yield a flexible empirical Bayes treatment of Dirichlet process mixtures. Such a treatment is useful in situations where smooth point estimates of G0 are of intrinsic interest, or where the structure of G0 cannot be conveniently modeled with the usual parametric prior families. Analysis of simulated and real-world datasets illustrates the robustness of this approach.

EM-Algorithm and Gibbs sampler are two useful Bayesian simulation methods for parameter estimation of finite normal mixture model. The EM-Algorithm is an iterative estimate of maximum likelihood for incomplete data problem. Gibbs sampler is an approach of generating random sample from a multivariate distribution. We introduce and derive Dempster EM-Algorithm for the two-component normal mixture models to get the iterative computation estimates, also use data augmentation and general Gibbs sampler to get the sample from posterior distribution under conjugate prior. The estimate results from both simulation methods under two-component normal mixture model with unknown mean parameters are compared and the connections and differences between both methods are represented. Data set from astronomy is used for comparison. Acknowledgement I would like to thank my supervisor Silvelyn Zwanzig for the patience, guidance and encouragement that she always gave to me, not only in the thesis, but also in the whole procedure of my statistics studying. I would also like to thank my friend Han Jun for the the assistances of LATEX, thank Alena for the data source, and thank my parents for the spiritual and substantial support and wholesouled love they gave me all my life. At last I would like to thank the department of mathematics of Uppsala University for giving me the opportunity to study. Contents 1

The Infinite Hidden Markov Model (IHMM) extends hidden Markov models to have a countably infinite number of hidden states (Beal et al., 2002; Teh et al., 2006). We present a generalization of this framework that introduces nearly block-diagonal structure in the transitions between the hidden states, where blocks correspond to “subbehaviors” exhibited by data sequences. In identifying such structure, the model classifies, or partitions, sequence data according to these sub-behaviors in an unsupervised way. We present an application of this model to artificial data, a video gesture classification task, and a musical theme labeling task, and show that components of the model can also be applied to graph segmentation. 1

Summary. We discuss functional clustering procedures for nested designs, where multiple curves are collected for each subject in the study. We start by considering the application of standard functional clustering tools to this problem, which leads to groupings based on the average profile for each subject. After discussing some of the shortcomings of this approach, we present a mixture model based on a generalization of the nested Dirichlet process that clusters subjects based on the distribution of their curves. By using mixtures of generalized Dirichlet processes, the model induces a much more flexible prior on the partition structure than other popular model-based clustering methods, allowing for different rates of introduction of new clusters as the number of observations increases. The methods are illustrated using hormone profiles from multiple menstrual cycles collected for women in the Early Pregnancy Study.

The immune response to vaccines and microbial pathogens is characterized by the spatial reorganization of leukocytes into microanatomical structures such as germinal centers and granulomas. Data on cellular organization is often provided by immunofluorescence histology, in which antibodies against specific molecules are conjugated (directly or indirectly) to fluorophores and used to stain thin sections of tissue for subsequent microscopic imaging. We have developed statistical tools to assist in the identification and quantitative characterization of cellular aggregates in immunofluorescent images. We model the spatial distribution of cells as a heterogeneous point process; the major inferential task then is the estimation of the Poisson intensity function underlying the point process. Note that this intensity function represents cellular density, not the fluorescence intensity itself. The intensity function is itself modeled as a flexible non-parametric Gaussian mixture model and provides the basis for the computation of statistics used to characterize the state of development of germinal centers and other cellular aggregates. We describe these methods and their efficient computational implementation and illustrate their use on highresolution images of lymph node sections stained for CD4, IgM, B220 and GL7. We identify and quantitatively characterize some of the major structural components of post-immunization lymph nodes such as B-cell follicles and germinal centers. 1

Bioequivalence trials are usually conducted to compare two or more formulations of a drug. Simultaneous assessment of bioequivalence on multiple endpoints is called multivariate bioequivalence. Despite the fact that some tests for multivariate bioequivalence are suggested, current practice usually involves univariate bioequivalence assessments ignoring the correlations between the endpoints such as AUC and Cmax. In this paper we develop a semiparametric Bayesian test for bioequivalence under multiple endpoints. Specifically, we show how the correlation between the endpoints can be incorporated in the analysis and how this correlation affects the inference. Resulting estimates and posterior probabilities “borrow strength” ’ from one another where the amount and direction of the strength borrowed are deter-