Some Aspects of Symmetric Gamma Process Mixtures

Naulet, Zacharie; Barat, Éric

doi:10.1214/17-ba1058

Cited by 3 publications

(3 citation statements)

References 18 publications

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“…Then, to use the RJMCMC algorithm, they obtained the location-scale parameterization for this distribution. Another study followed the location-scale parameterization for the RJMCMC method, namely research on the symmetry gamma distribution [59].…”

Section: Ng-rjmcmc Algorithmmentioning

confidence: 99%

On the Reversible Jump Markov Chain Monte Carlo (RJMCMC) Algorithm for Extreme Value Mixture Distribution as a Location-Scale Transformation of the Weibull Distribution

2021

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Data with a multimodal pattern can be analyzed using a mixture model. In a mixture model, the most important step is the determination of the number of mixture components, because finding the correct number of mixture components will reduce the error of the resulting model. In a Bayesian analysis, one method that can be used to determine the number of mixture components is the reversible jump Markov chain Monte Carlo (RJMCMC). The RJMCMC is used for distributions that have location and scale parameters or location-scale distribution, such as the Gaussian distribution family. In this research, we added an important step before beginning to use the RJMCMC method, namely the modification of the analyzed distribution into location-scale distribution. We called this the non-Gaussian RJMCMC (NG-RJMCMC) algorithm. The following steps are the same as for the RJMCMC. In this study, we applied it to the Weibull distribution. This will help many researchers in the field of survival analysis since most of the survival time distribution is Weibull. We transformed the Weibull distribution into a location-scale distribution, which is the extreme value (EV) type 1 (Gumbel-type for minima) distribution. Thus, for the mixture analysis, we call this EV-I mixture distribution. Based on the simulation results, we can conclude that the accuracy level is at minimum 95%. We also applied the EV-I mixture distribution and compared it with the Gaussian mixture distribution for enzyme, acidity, and galaxy datasets. Based on the Kullback–Leibler divergence (KLD) and visual observation, the EV-I mixture distribution has higher coverage than the Gaussian mixture distribution. We also applied it to our dengue hemorrhagic fever (DHF) data from eastern Surabaya, East Java, Indonesia. The estimation results show that the number of mixture components in the data is four; we also obtained the estimation results of the other parameters and labels for each observation. Based on the Kullback–Leibler divergence (KLD) and visual observation, for our data, the EV-I mixture distribution offers better coverage than the Gaussian mixture distribution.

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Section: Ng-rjmcmc Algorithmmentioning

confidence: 99%

On the Reversible Jump Markov Chain Monte Carlo (RJMCMC) Algorithm for Extreme Value Mixture Distribution as a Location-Scale Transformation of the Weibull Distribution

2021

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“…Our divide and conquer algorithm goes beyond density estimation problem as long as the prior of the relevant model consists of a Dirichlet distribution/process component, which is often seen along with models characterized by a(n) finite/infinite mixture of standard probability distributions. The popularity of such prior has risen in recent years with appearances in high dimensional normal means problem (Bhattacharya et al, 2015), multivariate cat-egorical data with dependency (Dunson and Xing, 2009), and nonlinear regression models (De Jonge et al, 2010;Naulet et al, 2018), just to name a few.…”

Section: Other Applicationsmentioning

confidence: 99%

A Divide and Conquer Algorithm of Bayesian Density Estimation

2020

Preprint

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Data sets for statistical analysis become extremely large even with some difficulty of being stored on one single machine. Even when the data can be stored in one machine, the computational cost would still be intimidating. We propose a divide and conquer solution to density estimation using Bayesian mixture modeling including the infinite mixture case. The methodology can be generalized to other application problems where a Bayesian mixture model is adopted. The proposed prior on each machine or subsample modifies the original prior on both mixing probabilities as well as on the rest of parameters in the distributions being mixed. The ultimate estimator is obtained by taking the average of the posterior samples corresponding to the proposed prior on each subset. Despite the tremendous reduction in time thanks to data splitting, the posterior contraction rate of the proposed estimator stays the same (up to a log factor) as that of the original prior when the data is analyzed as a whole. Simulation studies also justify the competency of the proposed method compared to the established WASP estimator in the finite dimension case. In addition, one of our simulations is performed in a shape constrained deconvolution context and reveals promising results. The application to a GWAS data set reveals the advantage over a naive method that uses the original prior.

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“…Simulation results. We use the algorithm of Naulet and Barat (2015) for simulating samples from posterior distributions of Gamma process mixtures. The base measure α on R 2 × [0, 2π] of the mixing Gamma process is taken as the independent product of a normal distribution on R 2 with covariance matrix diag(1/2, 1/2) and the uniform distribution on [0, 2π].…”

Section: Simulations Examplesmentioning

confidence: 99%

Bayesian nonparametric estimation for Quantum Homodyne Tomography

Naulet¹,

Barat²

2017

Electron. J. Statist.

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Abstract. We estimate the quantum state of a light beam from results of quantum homodyne tomography noisy measurements performed on identically prepared quantum systems. We propose two Bayesian nonparametric approaches. The first approach is based on mixture models and is illustrated through simulation examples. The second approach is based on random basis expansions. We study the theoretical performance of the second approach by quantifying the rate of contraction of the posterior distribution around the true quantum state in the L 2 metric.

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Some Aspects of Symmetric Gamma Process Mixtures

Cited by 3 publications

References 18 publications

On the Reversible Jump Markov Chain Monte Carlo (RJMCMC) Algorithm for Extreme Value Mixture Distribution as a Location-Scale Transformation of the Weibull Distribution

On the Reversible Jump Markov Chain Monte Carlo (RJMCMC) Algorithm for Extreme Value Mixture Distribution as a Location-Scale Transformation of the Weibull Distribution

A Divide and Conquer Algorithm of Bayesian Density Estimation

Bayesian nonparametric estimation for Quantum Homodyne Tomography

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