Compound Random Measures and their Use in Bayesian Non-Parametrics

Griffin, Jim E.; Leisen, Fabrizio

doi:10.1111/rssb.12176

Cited by 44 publications

(64 citation statements)

References 36 publications

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“…Here we consider the particular form

ρ false(normald w_{1}, \dots, normald w_{p} false) = \int_{0}^{\infty} w_{0}^{- p} F 2.047em2.047emtrue(\frac{d w_{1}}{w_{0}}, \dots, \frac{d w_{p}}{w_{0}} 2.047em2.047emtrue) ρ_{0} false(normald w_{0} false)

where

F (d β_{1}, \dots, d β_{p})

is some score probability distribution on

R_{+}^{p}

and ρ 0 is a base Lévy measure on

R_{+}

. The model defined by equations and (13) is a special case of the compound completely random measure (CCRM) model that was proposed by Griffin and Leisen (). It admits the following hierarchical construction, which makes interpretability, characterization of the conditionals and analysis of this class of models particularly easy.…”

Section: Sparse Graph Models With Overlapping Communitiesmentioning

confidence: 99%

“…We now give specific choices for the score distribution F and base Lévy measure ρ 0 , which lead to scalable inference algorithms. As in Griffin and Leisen (), we consider that F is a product of independent gamma distributions

F false(normald β_{1}, \dots, normald β_{p} false) = \prod_{k = 1}^{p} β_{k}^{a_{k} - 1} exp false(- b_{k} β_{k} false) \frac{b_{k}^{a_{k}}}{Γ (a_{k})} d β_{k}

where

a_{k} > 0

, b k >0, k =1,…, p . ρ 0 is set to be the mean measure of the jump part of a generalized gamma process (Hougaard, ; Brix, ), which has been extensively used in Bayesian non‐parametric models because of its generality, the interpretability of its parameters and its attractive conjugacy properties (James, ; Lijoi et al ., ; Saeedi and Bouchard‐Côté, ; Caron, ; Caron et al ., 2014).…”

Section: Sparse Graph Models With Overlapping Communitiesmentioning

confidence: 99%

“…We now give specific choices for the score distribution F and base Lévy measure ρ 0 , which lead to scalable inference algorithms. As in Griffin and Leisen (2017), we consider that F is a product of independent gamma distributions F.dβ 1 , :…”

Section: Specific Choices For F and ρmentioning

confidence: 99%

“…The construction of such multivariate point processes relies on vectors of completely random measures (CRMs) or equivalently multivariate subordinators. In particular, we build on the flexible though tractable construction that was recently introduced by Griffin and Leisen (2017).…”

Section: Introductionmentioning

confidence: 99%

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Exchangeable Random Measures for Sparse and Modular Graphs with Overlapping Communities

Todeschini

Caron²

2020

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary We propose a novel statistical model for sparse networks with overlapping community structure. The model is based on representing the graph as an exchangeable point process and naturally generalizes existing probabilistic models with overlapping block structure to the sparse regime. Our construction builds on vectors of completely random measures and has interpretable parameters, each node being assigned a vector representing its levels of affiliation to some latent communities. We develop methods for efficient simulation of this class of random graphs and for scalable posterior inference. We show that the approach proposed can recover interpretable structure of real world networks and can handle graphs with thousands of nodes and tens of thousands of edges.

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“…Here we consider the particular form

ρ false(normald w_{1}, \dots, normald w_{p} false) = \int_{0}^{\infty} w_{0}^{- p} F 2.047em2.047emtrue(\frac{d w_{1}}{w_{0}}, \dots, \frac{d w_{p}}{w_{0}} 2.047em2.047emtrue) ρ_{0} false(normald w_{0} false)

where

F (d β_{1}, \dots, d β_{p})

is some score probability distribution on

R_{+}^{p}

and ρ 0 is a base Lévy measure on

R_{+}

Section: Sparse Graph Models With Overlapping Communitiesmentioning

confidence: 99%

F false(normald β_{1}, \dots, normald β_{p} false) = \prod_{k = 1}^{p} β_{k}^{a_{k} - 1} exp false(- b_{k} β_{k} false) \frac{b_{k}^{a_{k}}}{Γ (a_{k})} d β_{k}

where

a_{k} > 0

Section: Sparse Graph Models With Overlapping Communitiesmentioning

confidence: 99%

Section: Specific Choices For F and ρmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Exchangeable Random Measures for Sparse and Modular Graphs with Overlapping Communities

Todeschini

Caron²

2020

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…Recent work such as Griffin et al . (), Griffin and Leisen () and Camerlenghi et al . () provides more flexible means to modelling the dependence between multiple samples than the classical models, but the aforementioned benefits of the Pólya tree remain even in view of these state of the art models.…”

Section: Modelmentioning

confidence: 94%

A Bayesian Hierarchical Model for Related Densities by using Pólya Trees

Christensen

2019

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary Bayesian hierarchical models are used to share information between related samples and to obtain more accurate estimates of sample level parameters, common structure and variation between samples. When the parameter of interest is the distribution or density of a continuous variable, a hierarchical model for continuous distributions is required. Various such models have been described in the literature using extensions of the Dirichlet process and related processes, typically as a distribution on the parameters of a mixing kernel. We propose a new hierarchical model based on the Pólya tree, which enables direct modelling of densities and enjoys some computational advantages over the Dirichlet process. The Pólya tree also enables more flexible modelling of the variation between samples, providing more informed shrinkage and permitting posterior inference on the dispersion function, which quantifies the variation between sample densities. We also show how the model can be extended to cluster samples in situations where the observed samples are believed to have been drawn from several latent populations.

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