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Über dieses Buch

This book reviews nonparametric Bayesian methods and models that have proven useful in the context of data analysis. Rather than providing an encyclopedic review of probability models, the book’s structure follows a data analysis perspective. As such, the chapters are organized by traditional data analysis problems. In selecting specific nonparametric models, simpler and more traditional models are favored over specialized ones.

The discussed methods are illustrated with a wealth of examples, including applications ranging from stylized examples to case studies from recent literature. The book also includes an extensive discussion of computational methods and details on their implementation. R code for many examples is included in online software pages.



Chapter 1. Introduction

We introduce the setup of nonparametric and semiparametric Bayesian models and inference.

Peter Müller, Fernando Andrés Quintana, Alejandro Jara, Tim Hanson

Chapter 2. Density Estimation: DP Models

We discuss the use of nonparametric Bayesian models in density estimation, arguably one of the most basic statistical inference problems. In this chapter we introduce the Dirichlet process prior and variations of it that are the by far most commonly used nonparametric Bayesian models used in this context. Variations include the Dirichlet process mixture and the finite Dirichlet process. One critical reason for the extensive use of these models is the availability of computation efficient methods for posterior simulation. We discuss several such methods.

Peter Müller, Fernando Andrés Quintana, Alejandro Jara, Tim Hanson

Chapter 3. Density Estimation: Models Beyond the DP

The ubiquitous use of Dirichlet process models should not discourage researchers from considering interesting features of alternative models. In particular, the Polya tree model turns out to be an attractive choice for some applications. In this chapter we discuss the use of the Polya tree prior and its variations for density estimation. We define the model, introduce computation efficient methods for posterior inference and identify relative advantages and limitations compared with Dirichlet process models.

Peter Müller, Fernando Andrés Quintana, Alejandro Jara, Tim Hanson

Chapter 4. Regression

Regression problems naturally call for nonparametric Bayesian methods when one wishes to relax restrictive parametric assumptions on the mean function, the residual distribution or both. We introduce suitable nonparametric Bayesian methods to facilitate such generalizations, including priors for random mean functions, the use of nonparametric density estimation for residual distributions and finally nonparametric Bayesian methods for fully nonparametric regression when both mean function and residual distribution are modeled nonparametrically. The latter includes approaches where the complete shape of the response distribution is allowed to change as a function of the predictors, which is also known as density regression. We introduce the popular dependent Dirichlet process model and several other alternatives.

Peter Müller, Fernando Andrés Quintana, Alejandro Jara, Tim Hanson

Chapter 5. Categorical Data

We discuss nonparametric Bayesian methods that are suitable for inference with binary, ordinal and general categorical data. Modeling for such data becomes particularly interesting in the presence of covariates, when non- and semi-parametric Bayesian models can generalize the link function in a generalized linear model setup, the regression on covariates or both. An important application arises in inference for diagnostic screening and related inference for ROC (receiver-operator characteristic) curves. We include some discussion of a rapidly growing literature on non-parametric Bayesian inference for ROC curves.

Peter Müller, Fernando Andrés Quintana, Alejandro Jara, Tim Hanson

Chapter 6. Survival Analysis

Inference for event time data is one of the most traditional applications of nonparametric Bayesian inference. For survival data, especially in biomedical applications, it is natural to focus on inference for detailed features of the survival function rather than only summaries like mean and variance. We extensively discuss semi- and nonparametric Bayesian methods for survival regression. Inference for such data has been traditionally dominated by the proportional hazards model. We review in detail nonparametric Bayesian alternatives which we introduce as natural generalizations of a parametric accelerated failure time model. We conclude with a discussion of three case studies.

Peter Müller, Fernando Andrés Quintana, Alejandro Jara, Tim Hanson

Chapter 7. Hierarchical Models

One of the great success stories of Bayesian methods in biostatistics is inference in hierarchical models. The model-based Bayesian approach allows for coherent propagation of uncertainties and borrowing of strength across submodels and more. In this chapter we discuss nonparametric Bayesian approaches in hierarchical models, including nonparametric priors on random effects distributions and extensions of such models across multiple related studies. Honest accounting for uncertainties becomes particularly important for applications to classification, when we use posterior predictive inference for a future experimental unit to estimate unknown membership in one of several subpopulations.

Peter Müller, Fernando Andrés Quintana, Alejandro Jara, Tim Hanson

Chapter 8. Clustering and Feature Allocation

An important byproduct of inference in discrete mixture models is an implied random partition of experimental units. In fact, such random partitions are the main inference targets for many recently published applications of nonparametric Bayesian discrete mixture models. In this chapter we systematically consider the use of nonparametric Bayesian priors for inference on such random partitions. Many scientific inference problems are formalized as the related, more general problem of feature allocation. That is, inference on possibly overlapping random subsets of experimental units. We introduce some examples from data analysis for bioinformatics data and introduce the Polya urn model, product partition models, model based clustering and the Indian buffet process prior.

Peter Müller, Fernando Andrés Quintana, Alejandro Jara, Tim Hanson

Chapter 9. Other Inference Problems and Conclusion

In this final chapter we briefly discuss some more specialized applications of nonparametric Bayesian inference, including the analysis of spatio-temporal data, model validation and causal inference. These themes are introduced to show by example the nature of the many application areas of nonparametric Bayesian inference that we did not include in earlier chapter.

Peter Müller, Fernando Andrés Quintana, Alejandro Jara, Tim Hanson


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