Generalized Gamma convolutions and complete monotonicity

Bondesson, Lennart

doi:10.1007/bf01277981

Cited by 20 publications

(19 citation statements)

References 17 publications

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“…We omit the proof of the fact that the inverted-beta distribution is a GGC, which is surprisingly involved; see Example 3.1 in Bondesson [1990]. The upshot of this result, however, is that the horseshoe prior can be represented as subordinated Brownian motion: the Lévy measure of the inverted-beta is concentrated on R + , and the corresponding independent-increments process therefore increases only by positive jumps.…”

Section: N Nmentioning

confidence: 99%

Shrink Globally, Act Locally: Sparse Bayesian Regularization and Prediction*

Polson¹,

Scott²

2011

Bayesian Statistics 9

288

344

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We study the classic problem of choosing a prior distribution for a location parameter β = (β 1 ,. .. , βp) as p grows large. First, we study the standard "global-local shrinkage" approach, based on scale mixtures of normals. Two theorems are presented which characterize certain desirable properties of shrinkage priors for sparse problems. Next, we review some recent results showing how Lévy processes can be used to generate infinite-dimensional versions of standard normal scale-mixture priors, along with new priors that have yet to be seriously studied in the literature. This approach provides an intuitive framework both for generating new regularization penalties and shrinkage rules, and for performing asymptotic analysis on existing models.

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Section: N Nmentioning

confidence: 99%

Shrink Globally, Act Locally: Sparse Bayesian Regularization and Prediction*

Polson¹,

Scott²

2011

Bayesian Statistics 9

288

344

View full text Add to dashboard Cite

show abstract

“…We omit the proof of the fact that the inverted beta distribution is self‐decomposable; see example 3.1 in Bondesson (1990). The consequence of this fact is that the horseshoe prior can be represented directly as subordinated Brownian motion.…”

Section: Priors and Penalties From Lévy Processesmentioning

confidence: 99%

Local Shrinkage Rules, Lévy Processes and Regularized Regression

Polson

Scott

2012

Journal of the Royal Statistical Society Series B: Statistical Methodology

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We use Lévy processes to generate joint prior distributions, and therefore penalty functions, for a location parameter β = (β 1 , . . . , β p ) as p grows large. This generalizes the class of local-global shrinkage rules based on scale mixtures of normals, illuminates new connections among disparate methods, and leads to new results for computing posterior means and modes under a wide class of priors. We extend this framework to large-scale regularized regression problems where p > n, and provide comparisons with other methodologies.

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“…is the density of a GGC, with K α,δ = √ 2δ/Γ(1 − α). Since δ ≤ 1/2 < 3/2, we see from Theorem 6.2 in [2] (and the Remark 6.1 thereafter) that this will be done as soon as…”

Section: Notationsmentioning

confidence: 97%

“…has CM derivative in w = v + 1/v for any fixed u, c > 0 (see [2], page 183), again Criterion 2 in [12], page 417, entails that the function ρ(uv)ρ(u/v) is CM in w, so that ρ is HCM.…”

Section: Related Hcm Densitiesmentioning

confidence: 99%

Further examples of GGC and HCM densities

Jedidi¹,

Simon²

2013

Bernoulli

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We display several examples of generalized gamma convoluted and hyperbolically completely monotone random variables related to positive $\alpha$-stable laws. We also obtain new factorizations for the latter, refining Kanter's and Pestana-Shanbhag-Sreehari's. These results give stronger credit to Bondesson's hypothesis that positive $\alpha$-stable densities are hyperbolically completely monotone whenever $\alpha\le1/2.$Comment: Published in at http://dx.doi.org/10.3150/12-BEJ431 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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Generalized Gamma convolutions and complete monotonicity

Abstract: Summary. Let

Cited by 20 publications

References 17 publications

Shrink Globally, Act Locally: Sparse Bayesian Regularization and Prediction*

Shrink Globally, Act Locally: Sparse Bayesian Regularization and Prediction*

Local Shrinkage Rules, Lévy Processes and Regularized Regression

Further examples of GGC and HCM densities

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