On the stick-breaking representation of normalized inverse Gaussian priors

Favaro, Stefano; Lijoi, Antonio; Prünster, Igor

doi:10.1093/biomet/ass023

Cited by 35 publications

(28 citation statements)

References 28 publications

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“…This extraordinarily clarifying perspective has inspired a number of nonparametric models (MacEachern, 1999, 2000; Hjort, 2000; Ishwaran and Zarepour, 2000; Ishwaran and James, 2001; Griffin and Steel, 2006; Dunson and Park, 2008; Chung and Dunson, 2009; Rodriguez and Dunson, 2011; Broderick et al, 2012), has provided insight into the properties of related models (Favaro et al, 2012; Teh et al, 2007; Thibaux and Jordan, 2007; Paisley et al, 2010), and has been used to develop efficient inference algorithms (Ishwaran and James, 2001; Blei and Jordan, 2006; Papaspiliopoulos and Roberts, 2008; Walker, 2007; Kalli et al, 2011). …”

Section: Equivalent Representationsmentioning

confidence: 99%

Mixture Models With a Prior on the Number of Components

Miller

Harrison

2017

Journal of the American Statistical Association

184

301

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A natural Bayesian approach for mixture models with an unknown number of components is to take the usual finite mixture model with symmetric Dirichlet weights, and put a prior on the number of components—that is, to use a mixture of finite mixtures (MFM). The most commonly-used method of inference for MFMs is reversible jump Markov chain Monte Carlo, but it can be nontrivial to design good reversible jump moves, especially in high-dimensional spaces. Meanwhile, there are samplers for Dirichlet process mixture (DPM) models that are relatively simple and are easily adapted to new applications. It turns out that, in fact, many of the essential properties of DPMs are also exhibited by MFMs—an exchangeable partition distribution, restaurant process, random measure representation, and stick-breaking representation—and crucially, the MFM analogues are simple enough that they can be used much like the corresponding DPM properties. Consequently, many of the powerful methods developed for inference in DPMs can be directly applied to MFMs as well; this simplifies the implementation of MFMs and can substantially improve mixing. We illustrate with real and simulated data, including high-dimensional gene expression data used to discriminate cancer subtypes.

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Section: Equivalent Representationsmentioning

confidence: 99%

Mixture Models With a Prior on the Number of Components

Miller

Harrison

2017

Journal of the American Statistical Association

184

301

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“…An important NRM is given by the normalized inverseGaussian N IG(c, G 0 ) process (Lijoi et al, 2005), which can be characterized as a stick-breaking process (Favaro et al, 2012), defined by the stick-breaking construction (12a-12d), after relaxing the i.i.d. assumption (12a), by allowing for dependence among the υ j distributions, with:…”

Section: Key Bnp Regression Modelsmentioning

confidence: 99%

“…given by equation (4) in Favaro et al (2012), and Eqs. 15b-15c refer to GIG and inverse-gamma (IG) distributions.…”

Section: Key Bnp Regression Modelsmentioning

confidence: 99%

A menu-driven software package of Bayesian nonparametric (and parametric) mixed models for regression analysis and density estimation

Karabatsos

2016

Behav Res

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Most of applied statistics involves regression analysis of data. In practice, it is important to specify a regression model that has minimal assumptions which are not violated by data, to ensure that statistical inferences from the model are informative and not misleading. This paper presents a stand-alone and menu-driven software package, Bayesian Regression: Nonparametric and Parametric Models, constructed from MATLAB Compiler. Currently, this package gives the user a choice from 83 Bayesian models for data analysis. They include 47 Bayesian nonparametric (BNP) infinite-mixture regression models; 5 BNP infinite-mixture models for density estimation; and 31 normal random effects models (HLMs), including normal linear models. Each of the 78 regression models handles either a continuous, binary, or ordinal dependent variable, and can handle multi-level (grouped) data. All 83 Bayesian models can handle the analysis of weighted observations (e.g., for meta-analysis), and the analysis of left-censored, right-censored, and/or interval-censored data. Each BNP infinite-mixture model has a mixture distribution assigned one of various BNP prior distributions, including priors defined by either the Dirichlet process, Pitman-Yor process (including the normalized stable process), beta (twoparameter) process, normalized inverse-Gaussian process, geometric weights prior, dependent Dirichlet process, or the dependent infinite-probits prior. The software user can mouse-click to select a Bayesian model and perform data George Karabatsos gkarabatsos1@gmail.com 1 University of Illinois, Chicago, IL, USA analysis via Markov chain Monte Carlo (MCMC) sampling. After the sampling completes, the software automatically opens text output that reports MCMC-based estimates of the model's posterior distribution and model predictive fit to the data. Additional text and/or graphical output can be generated by mouse-clicking other menu options. This includes output of MCMC convergence analyses, and estimates of the model's posterior predictive distribution, for selected functionals and values of covariates. The software is illustrated through the BNP regression analysis of real data.

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“…Moreover, (X j ) j ≥1 is a sequence of random variables, independent of (V j ) j ≥1 , and independent and identically distributed according to P 0 . See Favaro, Lijoi, and Prünster (2012) for a detailed analysis of the stick-breaking representation in Equation (13) with σ = 1/2.…”

Section: Sampling σ -Stable Poisson-kingman Mixture Modelsmentioning

confidence: 99%

Slice Sampling σ-Stable Poisson-Kingman Mixture Models

Favaro

Walker

2013

Journal of Computational and Graphical Statistics

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The article is concerned with the use of Markov chain Monte Carlo methods for posterior sampling in Bayesian nonparametric mixture models. In particular, we consider the problem of slice sampling mixture models for a large class of mixing measures generalizing the celebrated Dirichlet process. Such a class of measures, known in the literature as σ -stable Poisson-Kingman models, includes as special cases most of the discrete priors currently known in Bayesian nonparametrics, for example, the twoparameter Poisson-Dirichlet process and the normalized generalized Gamma process. The proposed approach is illustrated on some simulated data examples. This article has online supplementary material.

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On the stick-breaking representation of normalized inverse Gaussian priors

Cited by 35 publications

References 28 publications

Mixture Models With a Prior on the Number of Components

Mixture Models With a Prior on the Number of Components

A menu-driven software package of Bayesian nonparametric (and parametric) mixed models for regression analysis and density estimation

Slice Sampling σ-Stable Poisson-Kingman Mixture Models

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