On the stick-breaking representation of $\sigma$-stable Poisson-Kingman models

Favaro, Stefano; Lomelí, María; Nipoti, Bernardo; Teh, Yee Whye

doi:10.1214/14-ejs921

Cited by 6 publications

(5 citation statements)

References 28 publications

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“…It is well-known that the Dirichlet process with concentration parameter M has a stick-breaking representation of the form (32) with V i [188]. However, such representations for the general classes of random measure considered here are more recent developments, and the densities of the V i 's are quite complicated [52,50].…”

Section: Marginal Sampling Methodsmentioning

confidence: 97%

A review of uncertainty quantification for density estimation

McDonald

Campbell

2021

Statist. Surv.

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It is often useful to conduct inference for probability densities by constructing "plausible" sets in which the unknown density of given data may lie. Examples of such sets include pointwise intervals, simultaneous bands, or balls in a function space, and they may be frequentist or Bayesian in interpretation. For almost any density estimator, there are multiple approaches to inference available in the literature. Here we review such literature, providing a thorough overview of existing methods for density uncertainty quantification. The literature considered here comprises a spectrum from theoretical to practical ideas, and for some methods there is little commonality between these two extremes. After detailing some of the key concepts of nonparametric inference -the different types of "plausible" sets, and their interpretation and behaviour -we list the most prominent density estimators and the corresponding uncertainty quantification methods for each.

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Section: Marginal Sampling Methodsmentioning

confidence: 97%

A review of uncertainty quantification for density estimation

McDonald

Campbell

2021

Statist. Surv.

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“…An algorithm for slice sampling the sequence W was provided therein. Favaro et al [11] showed that, under certain assumptions on the parameter α, these sticks can be directly constructed with beta and gamma random variables.…”

Section: Stick-breaking Representationsmentioning

confidence: 99%

Gibbs-type Indian Buffet Processes

Heaukulani¹,

Roy

2020

Bayesian Anal.

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We investigate a class of feature allocation models that generalize the Indian buffet process and are parameterized by Gibbs-type random measures. Two existing classes are contained as special cases: the original two-parameter Indian buffet process, corresponding to the Dirichlet process, and the stable (or three-parameter) Indian buffet process, corresponding to the Pitman-Yor process. Asymptotic behavior of the Gibbs-type partitions, such as power laws holding for the number of latent clusters, translates into analogous characteristics for this class of Gibbs-type feature allocation models. Despite containing several different distinct subclasses, the properties of Gibbs-type partitions allow us to develop a black-box procedure for posterior inference within any subclass of models. Through numerical experiments, we compare and contrast a few of these subclasses and highlight the utility of varying power-law behaviors in the latent features.

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“…Fortunately, we can get an i.i.d. draw from the above due to an identity in distribution given by Favaro et al [8] for the usual stick breaking weights for any prior in this class such that σ " u v where u ă v are coprime integers. Then we just reparameterize it back to obtain the new size-biased weight, see Algorithm 3 in the supplementary material for details.…”

Section: Example Of Classes Of Poisson-kingman Priorsmentioning

confidence: 99%

“…The reason for this is that in the σ " 0.5 case there are readily available random number generators which do not increase the computational cost. In contrast, in the σ " 0.3 case, a rejection sampler method is needed every time a new size-biased weight is sampled which increases the computational cost, see Favaro et al [8] for details. Even so, in most cases, we outperform both marginal and conditional MCMC schemes in terms of running times and in all cases, in terms of ESS.…”

Section: Performance Assesssmentmentioning

confidence: 99%

A Marginal Sampler for σ-Stable Poisson–Kingman Mixture Models

Lomelí

Favaro

Teh

2017

Journal of Computational and Graphical Statistics

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We investigate the class of σ-stable Poisson-Kingman random probability We investigate the class of σ-stable Poisson-Kingman random probability measures (RPMs) in the context of Bayesian nonparametric mixture modeling. This is a large class of discrete RPMs which encompasses most of the popular discrete RPMs used in Bayesian nonparametrics, such as the Dirichlet process, Pitman-Yor process, the normalized inverse Gaussian process and the normalized generalized Gamma process. We show how certain sampling properties and marginal characterisations of σ-stable Poisson-Kingman RPMs can be usefully exploited for devising a Markov chain Monte Carlo (MCMC) algorithm for performing posterior inference with a Bayesian nonparametric mixture model. Specifically, we introduce a novel and efficient MCMC sampling scheme in an augmented space that has a fixed number of auxiliary variables per iteration. We apply our sampling scheme to a density estimation and clustering tasks with unidimensional and multidimensional datasets, and compare it against competing MCMC sampling schemes.

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On the stick-breaking representation of $\sigma$-stable Poisson-Kingman models

Cited by 6 publications

References 28 publications

A review of uncertainty quantification for density estimation

A review of uncertainty quantification for density estimation

Gibbs-type Indian Buffet Processes

A Marginal Sampler for σ-Stable Poisson–Kingman Mixture Models

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