Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems
TLDR
In this article, the conditional distribution of the random measure, given the observations, is no longer that of a simple Dirichlet process, but can be described as being a mixture of DirICHlet processes.Abstract:
process. This paper extends Ferguson's result to cases where the random measure is a mixing distribution for a parameter which determines the distribution from which observations are made. The conditional distribution of the random measure, given the observations, is no longer that of a simple Dirichlet process, but can be described as being a mixture of Dirichlet processes. This paper gives a formal definition for these mixtures and develops several theorems about their properties, the most important of which is a closure property for such mixtures. Formulas for computing the conditional distribution are derived and applications to problems in bio-assay, discrimination, regression, and mixing distributions are given.read more
Citations
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Hierarchical Dirichlet Processes
TL;DR: This work considers problems involving groups of data where each observation within a group is a draw from a mixture model and where it is desirable to share mixture components between groups, and considers a hierarchical model, specifically one in which the base measure for the childDirichlet processes is itself distributed according to a Dirichlet process.
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Bayesian Density Estimation and Inference Using Mixtures
Michael Escobar,Mike West +1 more
TL;DR: In this article, the authors describe and illustrate Bayesian inference in models for density estimation using mixtures of Dirichlet processes and show convergence results for a general class of normal mixture models.
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Markov Chain Sampling Methods for Dirichlet Process Mixture Models
TL;DR: In this article, Markov chain methods for sampling from the posterior distribution of a Dirichlet process mixture model are presented, and two new classes of methods are presented. But neither of these methods is suitable for handling general models with non-conjugate priors.
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Stochastic variational inference
TL;DR: Stochastic variational inference lets us apply complex Bayesian models to massive data sets, and it is shown that the Bayesian nonparametric topic model outperforms its parametric counterpart.
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Gibbs sampling methods for stick-breaking priors
TL;DR: Two general types of Gibbs samplers that can be used to fit posteriors of Bayesian hierarchical models based on stick-breaking priors are presented and the blocked Gibbs sampler, based on an entirely different approach that works by directly sampling values from the posterior of the random measure.
References
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Handbook of Mathematical Functions
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An Introduction to Probability Theory and Its Applications
David A. Freedman,William Feller +1 more
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A Bayesian Analysis of Some Nonparametric Problems
TL;DR: In this article, a class of prior distributions, called Dirichlet process priors, is proposed for nonparametric problems, for which treatment of many non-parametric statistical problems may be carried out, yielding results that are comparable to the classical theory.