On a class of $$\sigma $$ σ -stable Poisson–Kingman models and an effective marginalized sampler

Abstract

We investigate the use of a large class of discrete random probability measures, which is referred to as the class Q, in the context of Bayesian nonparametric mixture modeling. The class Q encompasses both the the two-parameter Poisson-Dirichlet process and the normalized generalized Gamma process, thus allowing us to comparatively study the inferential advantages of these two well-known nonparametric priors. Apart from a highly flexible parameterization, the distinguishing feature of the class Q is the availability of a tractable posterior distribution. This feature, in turn, leads to derive an efficient marginal MCMC algorithm for posterior sampling within the framework of mixture models. We demonstrate the efficacy of our modeling framework on both one-dimensional and multi-dimensional datasets. Keywords Bayesian nonparametrics • Normalized generalized Gamma process • Marginalized MCMC sampler • Mixture model • σ -Stable Poisson-Kingman model • Two parameter Poisson-Dirichlet process Electronic supplementary material The online version of this article (

A review of uncertainty quantification for density estimation

“…In their simulation studies, the credible intervals did a good job of capturing the true densities, covering them throughout the domains for all one-dimensional examples and at roughly 98% of domain points for their two-dimensional example. Favaro and Teh [54] and Favaro, Lomeli and Teh [53] devised marginal Gibbs samplers for NRMI's and a specific subclass of σ-stable Poisson-Kingman models, respectively. For both classes, their density draw computations appear 10 to be based on truncation, and marginalization of the distribution of G given θ and the auxiliary variables of the sampler.…”

“…Kottas [114] later used this approach in the context of survival analysis, as did Griffin [74] when comparing different approaches to hyperpriors in the Dirichlet process model. Such methodology is not typically used for more general random measure priors, despite relevant distributional results existing in the literature [54,53]. This is likely a computational matter: to directly sample the weights w of a random measure, it is typically necessary to employ a stick-breaking process, in which they are represented as…”

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

“…Marginal MCMC methods remove the infinite dimensionality of the problem by exploiting the tractable marginalization with respect to the Dirichlet process. See Escobar (1994), MacEachern (1994) and Escobar & West (1995) , Barrios et al (2013), Favaro & Teh (2013) and Favaro et al (2014) for details.…”

“…Recently, marginal and conditional MCMC methods have been developed under more general classes of mixing measures, such as stick-breaking RPMs and NRMs, among others. See Ishwaran & James (2001), , Barrios et al (2013), Favaro & Teh (2013) and Favaro et al (2014) for details.…”