| Description |
1 online resource : illustrations |
| Series |
Cambridge series in statistical and probabilistic mathematics |
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Cambridge series on statistical and probabilistic mathematics.
|
| Contents |
Cover; Half-title page; Series page; Title page; Copyright page; Dedication; Contents; Expanded Contents; Glossary of Symbols; Preface; 1 Introduction; 1.1 Motivation; 1.1.1 Classical versus Bayesian Nonparametrics; 1.1.2 Parametric versus Nonparametric Bayes; 1.2 Challenges of Bayesian Nonparametrics ; 1.2.1 Prior Construction; 1.2.2 Computation; 1.2.3 Asymptotic Behavior; 1.3 Priors, Posteriors and Bayes's Rule; 1.3.1 Absolute Continuity; 1.4 Historical Notes; 2 Priors on Function Spaces; 2.1 Random Basis Expansion; 2.2 Stochastic Processes; 2.2.1 Gaussian Processes |
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2.2.2 Increasing Processes2.3 Probability Densities; 2.3.1 Exponential Link Function; 2.3.2 Construction through Binning; 2.3.3 Mixtures; 2.3.4 Feller Approximation; 2.4 Nonparametric Normal Regression; 2.5 Nonparametric Binary Regression; 2.6 Nonparametric Poisson Regression; 2.7 Historical Notes; Problems; 3 Priors on Spaces of Probability Measures; 3.1 Random Measures; 3.1.1 Other Topologies; 3.2 Construction through a Stochastic Process; 3.3 Countable Sample Spaces; 3.3.1 Construction through Normalization; 3.3.2 Construction through Stick Breaking; 3.3.3 Countable Dirichlet Process |
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3.4 Construction through Structural Definitions3.4.1 Construction through a Distribution on a Dense Subset; 3.4.2 Construction through a Randomly Selected Discrete Set; 3.4.3 Construction through Random Rectangular Partitions; 3.4.4 Construction through Moments; 3.4.5 Construction through Quantiles; 3.4.6 Construction by Normalization; 3.5 Construction through a Tree; 3.6 Tail-Free Processes; 3.7 Pólya Tree Processes; 3.7.1 Relation with the Pólya Urn Scheme; 3.7.2 Mixtures of Pólya Tree Processes; 3.7.3 Partially Specified Pólya Tree; 3.7.4 Evenly Split Pólya Tree; 3.8 Historical Notes |
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Problems4 Dirichlet Processes; 4.1 Definition and Basic Properties; 4.1.1 Expectations, Variances and Co-Variances; 4.1.2 Self-Similarity; 4.1.3 Conjugacy; 4.1.4 Marginal and Conditional Distributions; 4.1.5 Number of Distinct Values; 4.2 Constructions; 4.2.1 Construction via a Stochastic Process; 4.2.2 Construction through Distribution Function; 4.2.3 Construction through a Gamma Process; 4.2.4 Construction through Pólya Urn Scheme; 4.2.5 Stick-Breaking Representation; 4.3 Further Properties; 4.3.1 Discreteness and Support; 4.3.2 Convergence; 4.3.3 Approximations |
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4.3.4 Mutual Singularity of Dirichlet Processes4.3.5 Tails of a Dirichlet Process; 4.3.6 Distribution of Median; 4.3.7 Distribution of Mean; 4.4 Characterizations; 4.5 Mixtures of Dirichlet Processes; 4.6 Modifications; 4.6.1 Invariant Dirichlet Process; 4.6.2 Constrained Dirichlet Process; 4.6.3 Penalized Dirichlet Process; 4.7 Bayesian Bootstrap; 4.8 Historical Notes; Problems; 5 Dirichlet Process Mixtures; 5.1 Dirichlet Process Mixtures; 5.2 MCMC Methods; 5.3 Variational Algorithm; 5.4 Predictive Recursion Deconvolution Algorithm; 5.5 Examples of Kernels; 5.6 Historical Notes; Problems; 6 Consistency: General Theory |
| Summary |
Explosive growth in computing power has made Bayesian methods for infinite-dimensional models - Bayesian nonparametrics - a nearly universal framework for inference, finding practical use in numerous subject areas. Written by leading researchers, this authoritative text draws on theoretical advances of the past twenty years to synthesize all aspects of Bayesian nonparametrics, from prior construction to computation and large sample behavior of posteriors. Because understanding the behavior of posteriors is critical to selecting priors that work, the large sample theory is developed systematically, illustrated by various examples of model and prior combinations. Precise sufficient conditions are given, with complete proofs, that ensure desirable posterior properties and behavior. Each chapter ends with historical notes and numerous exercises to deepen and consolidate the reader's understanding, making the book valuable for both graduate students and researchers in statistics and machine learning, as well as in application areas such as econometrics and biostatistics |
| Bibliography |
Includes bibliographical references and indexes |
| Notes |
Print version record |
| Subject |
Nonparametric statistics.
|
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Bayesian statistical decision theory.
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Bayes Theorem
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Statistics, Nonparametric
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Bayesian statistical decision theory
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Nonparametric statistics
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| Form |
Electronic book
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| Author |
Vaart, A. W. van der, author.
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| ISBN |
9781139029834 |
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1139029835 |
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