Construction Of Nonparametric Prior Distribution And Application

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 prior distributions we used most in treating nonparametric problems are those prior classes, eg: Dirichlet processes, Tailfree processes, neutral processes, Polya tree and so on. The Bayesian approach to statistical problems has been unsuccessful in treating nonparametic problems. This is due primarily to the limitations of prior distribution. It is necessary to consider whether there is a general method of construct prior distribution under some conditions. Based on two desirable properties of a prior distribution for nonparametric problems and some known prior distributions construction, some methods of construct prior distribution on countable sample spaces and uncountable sample spaces are introduced and given an algorithm to estimate the values of posterior means with a Dirichlet process prior.This paper does the work as following:1.Given methods of construction of prior distributions on countablesample spaces, i.e. construction via normalization and construction via stick-breaking2.Given methods of construction of prior distributions on uncount?able sample spaces, i.e. construct prior via binning,via increasing processes,via partitioning tree and so on.3.Discussed some properties and important facts of prior distribu?tion.4.Given an algorithm to estimate the values of posterior means witha Dirichlet process prior.