Nonparametric Bayesian Ananlysis Based On Dirichlet Process

The Bayesian approach to statistical problems, though fruitful in many ways, has been rather unsuccessful in treating nonparametric problems. This is due primarily to the difficulty in finding workable prior distributions on the parameter space, which in nonparametric problems is taken to be a set of probability distributions on a given sample space.Based on the paper of Ferguson in 1973, there are two desirable properties of a prior distribution for nonparametric problems:(1)The support of the prior distribution should be large-with respect to some suitable topology on the space of probability distributions on the sample space. This can assure the feasibility and universality of the prior, so we can find the best model for the distribution.(2) Posterior distributions given a sample of observations from the true probability should be manageable analytically. It requires the Posterior distributions have the same forms as the priors, or they are conjugate classes, or they can easily be computed.These properties are antagonistic in the sense that one may be obtained at the expense of the other. We usually broad a class of prior distributions in the sense of (1), for which (2) is realized by given in the sense of conjugate class.Refer to the papers in the past few decades, the method we used most is to follow the idea of Ferguson in 1973. The paper of Jayaram Sethuraman in 1994 is the best example of this idea, he gives a simple and new constructive definition of Dirichlet measures removing the restriction that the basic space should be R_k. In my paper, a unifying framework for Bayesian analysis in nonparametric setting is proposed. To this aim, a general class of nonparametric prior distributions on an arbitrary sample space is introduced. The general structure of posterior and predictive distributions are developed.This paper does the work as following:(1) Given the definition of simple Dirichlet process, proved that it is the prior distributions in nonparametric problems.(2) Discussed the properties of simple Dirichlet process, and given the means /variances/covariances/finite moments of the prior class.(3)Discussed the support of the prior class.(4)Computed the form of the Posterior...