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

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

doi:10.1080/10618600.2015.1110526

Cited by 8 publications

(7 citation statements)

References 54 publications

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“…Even for small N , this number is very large, which makes computation of the posterior intractable for the simplest choice of prior and likelihood. Thus, MCMC techniques are typically employed, such as the marginal samplers described by Neal [2000] with extensions in Favaro and Teh [2013] for normalized completely random measures and in Lomellí et al [2016] for σ-stable Poisson-Kingman models; the conditional samplers described in Ishwaran and James [2001], Papaspiliopoulos and Roberts [2008], or Kalli et al [2011], with extensions in Favaro and Teh [2013] for normalized completely random measures and in Favaro and Walker [2012] for σ-stable Poisson-Kingman models; or the recently introduced class of hybrid samplers for σ-stable Poisson-Kingman models in Lomellí et al [2015]. These algorithms produce approximate samples (c m ) M m=1 from the posterior (1).…”

Section: Bayesian Nonparametric Clusteringmentioning

confidence: 99%

Bayesian Cluster Analysis: Point Estimation and Credible Balls (with Discussion)

Wade¹,

Ghahramani²

2018

Bayesian Anal.

155

173

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Clustering is widely studied in statistics and machine learning, with applications in a variety of fields. As opposed to popular algorithms such as agglomerative hierarchical clustering or k-means which return a single clustering solution, Bayesian nonparametric models provide a posterior over the entire space of partitions, allowing one to assess statistical properties, such as uncertainty on the number of clusters. However, an important problem is how to summarize the posterior; the huge dimension of partition space and difficulties in visualizing it add to this problem. In a Bayesian analysis, the posterior of a real-valued parameter of interest is often summarized by reporting a point estimate such as the posterior mean along with 95% credible intervals to characterize uncertainty. In this paper, we extend these ideas to develop appropriate point estimates and credible sets to summarize the posterior of the clustering structure based on decision and information theoretic techniques.

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Section: Bayesian Nonparametric Clusteringmentioning

confidence: 99%

Bayesian Cluster Analysis: Point Estimation and Credible Balls (with Discussion)

Wade¹,

Ghahramani²

2018

Bayesian Anal.

155

173

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“…Although the PD(α, θ) class of models dominates the broad literature, there has been significant interest in the general class of Gibbs partitions. Here we note a few examples in [5,18,23,35,38,39,55,79]. Our exposition takes another viewpoint of this general class as we begin to describe next.…”

Section: Preliminaries On Poisson-kingman Distributions and Gibbs Par...mentioning

confidence: 99%

Gibbs Partitions, Riemann-Liouville Fractional Operators, Mittag-Leffler Functions, and Fragmentations Derived From Stable Subordinators

James

Lau

2018

Preprint

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Pitman [75] (and subsequently Gnedin and Pitman [31]) showed that a large class of random partitions of the integers derived from a stable subordinator of index α ∈ (0, 1) have infinite Gibbs (product) structure as a characterizing feature. The most notable case are random partitions derived from the two-parameter Poisson-Dirichlet distribution, PD(α, θ), which are induced by mixing over variables with generalized Mittag-Leffler distributions, denoted by ML(α, θ). Our aim in this work is to provide indications on the utility of the wider class of Gibbs partitions as it relates to a study of Riemann-Liouville fractional integrals and size-biased sampling, decompositions of special functions, and its potential use in the understanding of various constructions of more exotic processes. We provide novel characterizations of general laws associated with two nested families of PD(α, θ) mass partitions that are constructed from notable fragmentation operations described in Dong, Goldschmidt and Martin [27] and Pitman [73], respectively. These operations are known to be related in distribution to various constructions of discrete random trees/graphs in [n], and their scaling limits, such as stable trees. A centerpiece of our work are results related to Mittag-Leffler functions, which play a key role in fractional calculus and are otherwise Laplace transforms of the ML(α, θ) variables. Notably, this * Supported in part by the grant RGC-GRF 16300217 of the HKSAR. 1 leads to an interpretation of PD(α, θ) laws within a mixed Poisson waiting time framework based on ML(α, θ) variables, which suggests connections to recent construction of Pólya urn models with random immigration by Peköz, Röllin and Ross [65]. Simplifications in the Brownian case are highlighted.

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“…Gibbs-type species sampling models have been also applied in the context of mixture modeling, thus generalizing the seminal work by Lo [47]. See, e.g., Ishwaran and James [34], Lijoi et al [41], Lijoi et al [42], Favaro and Walker [23] and Lomeli et al [49]. While maintaining the same computational tractability of the Dirichlet process mixture model, the availability of the additional parameter α allows for a better control of the clustering behaviour.…”

Section: A Brief Review Of Gibbs-type Priorsmentioning

confidence: 99%

Sufficientness Postulates for Gibbs-Type Priors and Hierarchical Generalizations

Bacallado¹,

Battiston²,

Favaro³

et al. 2017

Statist. Sci.

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A fundamental problem in Bayesian nonparametrics consists of selecting a prior distribution by assuming that the corresponding predictive probabilities obey certain properties. An early discussion of such a problem, although in a parametric framework, dates back to the seminal work by English philosopher W. E. Johnson, who introduced a noteworthy characterization for the predictive probabilities of the symmetric Dirichlet prior distribution. This is typically referred to as Johnson's "sufficientness" postulate. In this paper we review some nonparametric generalizations of Johnson's postulate for a class of nonparametric priors known as species sampling models. In particular we revisit and discuss the "sufficientness" postulate for the two parameter Poisson-Dirichlet prior within the more general framework of Gibbstype priors and their hierarchical generalizations.

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

Cited by 8 publications

References 54 publications

Bayesian Cluster Analysis: Point Estimation and Credible Balls (with Discussion)

Bayesian Cluster Analysis: Point Estimation and Credible Balls (with Discussion)

Gibbs Partitions, Riemann-Liouville Fractional Operators, Mittag-Leffler Functions, and Fragmentations Derived From Stable Subordinators

Sufficientness Postulates for Gibbs-Type Priors and Hierarchical Generalizations

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