Bayesian Theory For Some Statistical Problems

The issues at the very core of theories and applications of Bayesian statistics are the choice of a prior distribution and the calculation of the posterior distribution.Working around this core,this dissertation studies the following three aspects.When a prior is given for the parameter which is to be estimated,starting from the maximum likelihood estimator for this parameter,a random weighting method is pro-posed for an approximation of the posterior distribution,which does not require iteration and which has great computational advantages.It is proved that a convergence rate of o(n^-1/2)can be achieved by applying this method,which is much better than a usual normal approximation.In simulations decent performances are exhibited under various priors,the conclusions of the theorems are verified and the role that a prior plays is ex-plained.This subject emphasizes the importance of calculating posterior distributions.Binary response data is often modeled via a probit or a logistic model.Bayesian variable selection methods even workable when p>>n are proposed for both of these,in essence using shrinking and diffusing priors.For the first,due to the underlying normal structure,a data augmentation method is applied to transform the model into a linear one.For the second,a class of P?olya-Gamma distributions is used to approximate a logistic distribution.Improved performances of these new methods are found through compar-isons with common methods.A contrast between these two models is also provided,answering the question of how to make a choice with real data.This subject stresses the significance of choosing prior distributions.When seeking full Bayesian inference of a proportional odds model for right censored data,a process neutral to the right is used as the prior for the nonparametric component,and an absolutely continuous finite-dimensional prior is placed on the parametric component.Expressed in terms of the L?evy measure of a random distribution,the exact forms of the marginal posterior distributions of the regression coefficient and the baseline cumulative distribution function are obtained.Under some regularity conditions,it is proved that the posterior distribution of the regression coefficient with a constant prior is proper,which serves as a theoretical basis for the Bayesian analysis of proportional odds models.This subject advances to the consideration of how to define a prior and calculate a posterior for an unknown function.The three main contributions made by this dissertation are as follows:·Using a frequentist random weighting method to solve the Bayesian problem of approximating a posterior distribution is progressive in the philosophical sense;·Adopting a data-driven prior may be against the Bayesian philosophy,but the excellent results found using it make it deserve to be recommended;·The subject of Bayesian analysis is extended from a common random variable to an unknown function,showing a constructive effect on this new study field.