Distributional results for means of normalized random measures with independent increments

Regazzini, Eugenio; Lijoi, Antonio; Prünster, Igor

doi:10.1214/aos/1051027881

Cited by 195 publications

(159 citation statements)

References 22 publications

Supporting

Mentioning

157

Contrasting

Unclassified

Order By: Relevance

“…This model formulation, however, does not allow the exchange of information among the vectors of observed data, since independent Dirichlet process priors are used for each of the n samples. Cremaschi et al (2019) improve on this model by using a hierarchical construction based on a more flexible class of nonparametric prior distributions, known as normalized completely random measures (NormCRMs), first introduced by Regazzini et al (2003). Furthermore, Bhadra et al (2018) allow extensions to mixtures of continuous and discrete-valued (binary or ordinal) nodes through a latent variable framework for inferring conditional independence structures.…”

Section: Robust Graphical Modelsmentioning

confidence: 99%

Bayesian graphical models for modern biological applications

Baladandayuthapani

Vannucci

et al. 2021

Stat Methods Appl

View full text Add to dashboard Cite

Graphical models are powerful tools that are regularly used to investigate complex dependence structures in high-throughput biomedical datasets. They allow for holistic, systems-level view of the various biological processes, for intuitive and rigorous understanding and interpretations. In the context of large networks, Bayesian approaches are particularly suitable because it encourages sparsity of the graphs, incorporate prior information, and most importantly account for uncertainty in the graph structure. These features are particularly important in applications with limited sample size, including genomics and imaging studies. In this paper, we review several recently developed techniques for the analysis of large networks under non-standard settings, including but not limited to, multiple graphs for data observed from multiple related subgroups, graphical regression approaches used for the analysis of networks that change with covariates, and other complex sampling and structural settings. We also illustrate the practical utility of some of these methods using examples in cancer genomics and neuroimaging.

show abstract

Section: Robust Graphical Modelsmentioning

confidence: 99%

Bayesian graphical models for modern biological applications

Baladandayuthapani

Vannucci

et al. 2021

Stat Methods Appl

View full text Add to dashboard Cite

show abstract

“…Favaro et al (2016) present the stick-breaking representation of homogeneous normalized random measures with independent increments (hNRMIs) (see e.g. Regazzini et al, 2003 for more details of NRMIs), which include the normalized generalized gamma process and the generalized Dirichlet process, two widely used priors in Bayesian nonparametric statistics.…”

Section: Introductionmentioning

confidence: 99%

Functional Central Limit Theorems for Stick-Breaking Priors

Zhang

2022

Bayesian Anal.

View full text Add to dashboard Cite

We obtain the strong law of large numbers, Glivenko-Cantelli theorem, central limit theorem, functional central limit theorem for various Bayesian nonparametric priors which include the stick-breaking process with general stickbreaking weights, the two-parameter Poisson-Dirichlet process, the normalized inverse Gaussian process, the normalized generalized gamma process, and the generalized Dirichlet process. For the stick-breaking process with general stick-breaking weights, we introduce two general conditions such that the central limit theorem and functional central limit theorem hold. Except in the case of the generalized Dirichlet process, since the finite dimensional distributions of these processes are either hard to obtain or are complicated to use even they are available, we use the method of moments to obtain the convergence results. For the generalized Dirichlet process we use its marginal distributions to obtain the asymptotics although the computations are highly technical.

show abstract

“…We propose to use a more flexible class of nonparametric prior distributions, known as normalized completely random measures (NormCRMs), and consider a hierarchical construction where the nonparametric priors for the divisors are conditionally independent, given their centering measure, which is itself a completely random measure. NormCRMs were first introduced by Regazzini et al (2003) with the name of Normalized Random Measures with Independent increments (NRMI), and subsequently studied by several researchers in statistics and machine learning (James et al, 2009;Lijoi and Prünster, 2010;Caron and Fox, 2017). One of the most commonly used measures in this class is the Normalized Generalized Gamma (NGG) process (Lijoi et al, 2007).…”

Section: Introductionmentioning

confidence: 99%

Hierarchical Normalized Completely Random Measures for Robust Graphical Modeling

Cremaschi¹,

Argiento²,

Shoemaker³

et al. 2019

Bayesian Anal.

View full text Add to dashboard Cite

Gaussian graphical models are useful tools for exploring network structures in multivariate normal data. In this paper we are interested in situations where data show departures from Gaussianity, therefore requiring alternative modeling distributions. The multivariate t-distribution, obtained by dividing each component of the data vector by a gamma random variable, is a straightforward generalization to accommodate deviations from normality such as heavy tails. Since different groups of variables may be contaminated to a different extent, Finegold and Drton (2014) introduced the Dirichlet t-distribution, where the divisors are clustered using a Dirichlet process. In this work, we consider a more general class of nonparametric distributions as the prior on the divisor terms, namely the class of normalized completely random measures (NormCRMs). To improve the effectiveness of the clustering, we propose modeling the dependence among the divisors through a nonparametric hierarchical structure, which allows for the sharing of parameters across the samples in the data set. This desirable feature enables us to cluster together different components of multivariate data in a parsimonious way. We demonstrate through simulations that this approach provides accurate graphical model inference, and apply it to a case study examining the dependence structure in radiomics data derived from The Cancer Imaging Atlas.

show abstract

Distributional results for means of normalized random measures with independent increments

Cited by 195 publications

References 22 publications

Bayesian graphical models for modern biological applications

Bayesian graphical models for modern biological applications

Functional Central Limit Theorems for Stick-Breaking Priors

Hierarchical Normalized Completely Random Measures for Robust Graphical Modeling

Contact Info

Product

Resources

About