East China Normal University Several Applications Of Bayesian Nonparametric Priors

Bayesian nonparanetrics is a relatively young,yet fast growing field of statistics,in which it not only produces a large number er of theoretical achievements but also widely ap-plies itself in various substantive fields and directions.Compared to traditional frequency nonparametric statistics,Bayesian nonparametrics is based on its posterior or predictive distribution for statistical inference in Bayesian framework.And compared to parametric Bayesian,they build such prior distributions that sit on larger spaces of distributions of interest rather than spaces of parameters of parametrically specified distributionsAlthough Bayesian non-parametrics may face a lot of challenges especially for the calculation of the problem.A long time ago,people only focused on the theory of Bayesian nonparanietrics due to the difficulty of calculation.Fortunately,Baysian nonparametrics are applied in the study of practical problems and analyse more complex data with the recent development of computer and numerical analysis such as MCMC.The mostly common used priors of Bayesian nonparametrics include Stick-breaking process prior Dirichlet process prior,Pitman-Yor process prior and Polya tree so onIn this dissertation,we mainly discuss the general structure of the Stick-breaking process priors and their theory properties,experience ratemaking under Dirichlet process priors,Bayesian ratemaking with common effects modeled by mixture of Polya,tree process and The Estimation of Copulas for M ixture of Finite Multivariate Polya trees,respectively The contents of this dissertation are as followsFirstly,we provide a comprehensive review on Bayesian nonparametrics in Chapter 1 including why we apply Bayesian nonparametrics,three different Bayesian nonparametric priors,a brief history of Bayesian nonparametrics,its abundant theoretical achievements and applications.We also discuss the computational issues,future research directions and potential challenges to Bayesian nonparametrics,via recalling a series of literatures from numerous statisticiansIn chapter 2,We discuss two novel classes for Bayesian nonparametric priors based on stick-breaking construction proposed for another definition of Dirichlet process priors by(Sethuraman 1994)where the constructions of the stick-breaking ratios are assumed to follow identical distribution or different distribution families including two known pro-cesses:Dirichlet process and Poisson-Dirichlet process.Some theoretical properties of Bayesian nonparametric priors are described and these priors are extremely flexible,al-lowing us to generate a great variety models.Three different conditional methods for these general stick-breaking processes are considered for posterior computation and they are illustrated using a simulated example.The chapter 3 investigates Bayes premium under different various error functions in which claims can be modeled by certain parametric family of distributions and Dirichlet process mixtures are employed to model the distributions of the parameters.There are two advantages:to produce exact Bayesian experience premiums for a class of premium principles generated from generic error functions and,at the same time,provide robust and flexible ways to avoid possible bias caused by traditionally used priors such as nonin-formative priors or conjugate priors.In this chapter,the Bayesian experience ratemaking under Dirichlet process mixture models are investigated and due to the lack of analytical forms of the conditional expectations of the quantities concerned,the Gibbs sampling schemes are designed for the purpose of approximations.The purpose of chapter 4 is to investigate the typical problem of experience ratemak-ing for Polya tree priors in the framework of Bayesian non-parametrics with various error functions.The usual routine for Bayes method is to assume that the data follow a per-fectly specified parametric density and the parameter is assigned a specified structure function to reflect the prior information.But the information of analysts is often not sufficient to provide complete specifications for the priors especially in practise.Thus,it is naturally to assume the prior density is generated by the Polya tree process,which belongs to Bayesian nonparameteric analysis method.The computation of posterior dis-tribution or conditional expectation of the concerned parameters would be very complex and thus numerical result should be exploited to approximate for obtaining Bayes pre-mium through MCMC method.Using finite Polya tree priors for risk parameter,we demonstrate advantages over parametric Bayesian modeling in a simulated example.Chapter 5 provides a Bayesian nonparametric approach to the estimation of Copula function which can fully characterize the dependence of multiple variables,i.e.functions linking joint distributions to their univariate margins.We do this by employing multi-variate Polya tree models where they can handle nonstandard features in the data.We address the computational challenges using Markov Chain Monte Carlo method especial-ly when we apply multivariate Polya tree priors to model higher dimension data sets.A simulation study with Guassian distribution is performed to assess the performance of the proposed method containing two cases,independent and comonotonic.The results based on real data will be reported to analysis the dependence existing among them where the data come from Shanghai Composite Index and Shenzhen index.