The local Dirichlet process (original) (raw)
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Nonparametric empirical Bayes for the Dirichlet process mixture model
2006
Abstract 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 arrive. However, analyses based on the Dirichlet process prior are sensitive to the choice of the parameters, including an infinite-dimensional distributional parameter G 0. Most previous applications have either fixed G 0 as a member of a parametric family or treated G 0 in a Bayesian fashion, using parametric prior specifications.
The Dependent Dirichlet Process and Related Models
2020
Standard regression approaches assume that some finite number of the response distribution characteristics, such as location and scale, change as a (parametric or nonparametric) function of predictors. However, it is not always appropriate to assume a location/scale representation, where the error distribution has unchanging shape over the predictor space. In fact, it often happens in applied research that the distribution of responses under study changes with predictors in ways that cannot be reasonably represented by a finite dimensional functional form. This can seriously affect the answers to the scientific questions of interest, and therefore more general approaches are indeed needed. This gives rise to the study of fully nonparametric regression models. We review some of the main Bayesian approaches that have been employed to define probability models where the complete response distribution may vary flexibly with predictors. We focus on developments based on modifications of ...
2002
Widely used parametric generalizedlinear models are, unfortunately,a somewhat limited class of speci cations. Nonparametric aspects are often introduced to enrich this class, resulting in semiparametricmodels. Focusing on single or k-sample problems, many classical nonparametricapproachesare limited to hypothesis testing. Those that allow estimation are limited to certain functionals of the underlying distributions. Moreover, the associated inference often relies upon asymptotics when nonparametric speci cations are often most appealing for smaller sample sizes. Bayesian nonparametric approaches avoid asymptotics but have, to date, been limited in the range of inference. Working with Dirichlet process priors, we overcome the limitations of existing simulation-based model tting approaches which yield inference that is con ned to posterior moments of linear functionals of the population distribution. This article provides a computational approach to obtain the entire posterior distribution for more general functionals. We illustrate with three applications: investigation of extreme value distributions associated with a single population, comparison of medians in a k-sample problem, and comparison of survival times from different populations under fairly heavy censoring.
Electronic Journal of Statistics, 2016
In this article we propose novel Bayesian nonparametric methods using Dirichlet Process Mixture (DPM) models for detecting pairwise dependence between random variables while accounting for uncertainty in the form of the underlying distributions. A key criteria is that the procedures should scale to large data sets. In this regard we find that the formal calculation of the Bayes factor for a dependent-vs.-independent DPM joint probability measure is not feasible computationally. To address this we present Bayesian diagnostic measures for characterising evidence against a "null model" of pairwise independence. In simulation studies, as well as for a real data analysis, we show that our approach provides a useful tool for the exploratory nonparametric Bayesian analysis of large multivariate data sets.
Variational inference for Dirichlet process mixtures
2006
Abstract Dirichlet process (DP) mixture models are the cornerstone of nonparametric Bayesian statistics, and the development of Monte-Carlo Markov chain (MCMC) sampling methods for DP mixtures has enabled the application of nonparametric Bayesian methods to a variety of practical data analysis problems. However, MCMC sampling can be prohibitively slow, and it is important to explore alternatives.
Theory and computations for the Dirichlet process and related models: An overview
International Journal of Approximate Reasoning, 2017
Data analysis sometimes requires the relaxation of parametric assumptions in order to gain modeling flexibility and robustness against mis-specification of the probability model. In the Bayesian context, this is accomplished by placing a prior distribution on an infinitedimensional space, referred to as Bayesian nonparametric models. We provide an overview on the most popular Bayesian nonparametric models for probability distributions and for collections of predictor-dependent probability distributions. The intention of is not to be complete or exhaustive, but rather to touch on areas of interest for the practical use of the priors in the context of a hierarchical model. We give an overview covering the main properties of the basic models and the algorithms for fitting them.
A Dirichlet Process Prior Approach for Covariate Selection
Entropy
The variable selection problem in general, and specifically for the ordinary linear regression model, is considered in the setup in which the number of covariates is large enough to prevent the exploration of all possible models. In this context, Gibbs-sampling is needed to perform stochastic model exploration to estimate, for instance, the model inclusion probability. We show that under a Bayesian non-parametric prior model for analyzing Gibbs-sampling output, the usual empirical estimator is just the asymptotic version of the expected posterior inclusion probability given the simulation output from Gibbs-sampling. Other posterior conditional estimators of inclusion probabilities can also be considered as related to the latent probabilities distributions on the model space which can be sampled given the observed Gibbs-sampling output. This paper will also compare, in this large model space setup the conventional prior approach against the non-local prior approach used to define the...
Improving prediction from Dirichlet process mixtures via enrichment
Flexible covariate-dependent density estimation can be achieved by modelling the joint density of the response and covariates as a Dirichlet process mixture. An appealing aspect of this approach is that computations are relatively easy. In this paper, we examine the predictive performance of these models with an increasing number of covariates. Even for a moderate number of covariates, we find that the likelihood for x tends to dominate the posterior of the latent random partition, degrading the predictive performance of the model. To overcome this, we suggest using a different nonparametric prior, namely an enriched Dirichlet process. Our proposal maintains a simple allocation rule, so that computations remain relatively simple. Advantages are shown through both predictive equations and examples, including an application to diagnosis Alzheimer's disease.
Journal of the American Statistical Association, 2008
In multicenter studies, subjects in different centers may have different outcome distributions. This article is motivated by the problem of nonparametric modeling of these distributions, borrowing information across centers while also allowing centers to be clustered. Starting with a stickbreaking representation of the Dirichlet process (DP), we replace the random atoms with random probability measures drawn from a DP. This results in a nested Dirichlet process (nDP) prior, which can be placed on the collection of distributions for the different centers, with centers drawn from the same DP component automatically clustered together. Theoretical properties are discussed, and an efficient MCMC algorithm is developed for computation. The methods are illustrated using a simulation study and an application to quality of care in US hospitals.