Bayesian Predictive Distribution for a Negative Binomial Model

Hamura, Yasuyuki; Kubokawa, Tatsuya

doi:10.3103/s1066530719010010

Cited by 8 publications

(4 citation statements)

References 15 publications

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“…One important direction of future research is the generalization of the present results for nonhomogeneous Poisson process models to those for other stochastic process models such as nonhomogeneous negative binomial process models. For finite dimensional models, [15], [16] extended the theory for finite dimensional Poisson models to finite dimensional negative binomial models and negative multinomial models. These generalizations require techniques not used in the theory for the Poisson models.…”

Section: Discussionmentioning

confidence: 99%

Shrinkage Priors for Nonparametric Bayesian Prediction of Nonhomogeneous Poisson Processes

Komaki

2021

IEEE Trans. Inform. Theory

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We consider nonparametric Bayesian estimation and prediction for nonhomogeneous Poisson process models with unknown intensity functions. We propose a class of improper priors for intensity functions. Nonparametric Bayesian inference with kernel mixture based on the class improper priors is shown to be useful, although improper priors have not been widely used for nonparametric Bayes problems. Several theorems corresponding to those for finite-dimensional independent Poisson models hold for nonhomogeneous Poisson process models with infinite-dimensional parameter spaces. Bayesian estimation and prediction based on the improper priors are shown to be admissible under the Kullback-Leibler loss. Numerical methods for Bayesian inference based on the priors are investigated.

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Section: Discussionmentioning

confidence: 99%

Shrinkage Priors for Nonparametric Bayesian Prediction of Nonhomogeneous Poisson Processes

Komaki

2021

IEEE Trans. Inform. Theory

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“…Similar results in the context of predictive density estimation under the Kullback-Leibler (KL) divergence have also been obtained. Fourdrinier et al (2011), Hamura and Kubokawa (2019), and Hamura and Kubokawa (2020) treated the normal, negative binomial, and Poisson cases by using identies which relate Bayesian predictive density estimation to Bayesian point estimation. L'Moudden et al (2017) and Hamura and Kubokawa (2021) considered the gamma and exponential cases directly.…”

Section: Introductionmentioning

confidence: 99%

Bayesian Point Estimation and Predictive Density Estimation for the Binomial Distribution with a Restricted Probability Parameter

Hamura

2021

Preprint

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In this paper, we consider Bayesian point estimation and predictive density estimation in the binomial case. After presenting preliminary results on these problems, we compare the risk functions of the Bayes estimators based on the truncated and untruncated beta priors and obtain dominance conditions when the probability parameter is less than or equal to a known constant. The case where there are both a lower bound restriction and an upper bound restriction is also treated. Then our problems are shown to be related to similar problems in the Poisson case. Finally, numerical studies are presented.

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“…Even in the Poisson case, it was only after the work of Komaki (2015) that many Bayesian shrinkage estimators were shown to dominate usual estimators in the presence of unbalanced sample sizes Kubokawa (2019b, 2020c)). Third, while theoretical properties of Bayesian predictive densities for Poisson models have been investigated in several papers as mentioned earlier, relatively few researchers (Komaki (2012), Hamura and Kubokawa (2019a)) have considered predictive density estimation for other discrete exponential families. In this paper, we treat these three issues when considering Bayesian estimators and predictive density estimators based on negative multinomial observations in unbalanced settings.…”

Section: Introductionmentioning

confidence: 99%

“…Whereas Komaki (2012) investigated asymptotic properties of Bayesian predictive densities for future multinomial observations based on current multinomial observations, the sample space is not a finite set in our setting and we investigate finite sample properties of Bayesian predictive densities. Although Hamura and Kubokawa (2019a) considered Bayesian predictive densities for a negative binomial model, where a future observation also is negative binomial and can take on an infinite number of values, they did not treat the problem of estimating the joint predictive density of multiple negative binomial observations.…”

Section: Introductionmentioning

confidence: 99%

Bayesian Shrinkage Approaches to Unbalanced Problems of Estimation and Prediction on the Basis of Negative Multinomial Samples

Hamura

2020

Preprint

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In this paper, we treat estimation and prediction problems where negative multinomial variables are observed and in particular consider unbalanced settings. First, the problem of estimating multiple negative multinomial parameter vectors under the standardized squared error loss is treated and a new empirical Bayes estimator which dominates the UMVU estimator under suitable conditions is derived. Second, we consider estimation of the joint predictive density of several multinomial tables under the Kullback-Leibler divergence and obtain a sufficient condition under which the Bayesian predictive density with respect to a hierarchical shrinkage prior dominates the Bayesian predictive density with respect to the Jeffreys prior. Third, our proposed Bayesian estimator and predictive density give risk improvements in simulations. Finally, the problem of estimating the joint predictive density of negative multinomial variables is discussed.

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Bayesian Predictive Distribution for a Negative Binomial Model

Cited by 8 publications

References 15 publications

Shrinkage Priors for Nonparametric Bayesian Prediction of Nonhomogeneous Poisson Processes

Shrinkage Priors for Nonparametric Bayesian Prediction of Nonhomogeneous Poisson Processes

Bayesian Point Estimation and Predictive Density Estimation for the Binomial Distribution with a Restricted Probability Parameter

Bayesian Shrinkage Approaches to Unbalanced Problems of Estimation and Prediction on the Basis of Negative Multinomial Samples

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