Polya tree distributions for statistical modeling of censored data

Andrew A. Neath

doi:10.1155/s1173912603000166

Abstract

Polya tree distributions extend the idea of the Dirichlet process as a prior for Bayesian nonparametric problems. Finite dimensional distributions are defined through conditional probabilities in P. This allows for a specification of prior information which carries greater weight where it is deemed appropriate according to the choice of a partition of the sample space. Muliere and Walker[7] construct a partition so that the posterior from right censored data is also a Polya tree. A point of contention is that the specification of the prior is partially dependent on the data. In general, the posterior from censored data will be a mixture of Polya trees. This paper will present a straightforward method for determining the mixing distribution.

Highlights

In a nonparametric statistical model, the unknown of interest is the probability measure responsible for generating the data at hand
We say that the probability measure P follows a Dirichlet process with parameter α if for every measurable partition (B1, . . . , Bk) of, (P (B1), ..., P (Bk)) follows a Dirichlet distribution with parameter vector (α(B1), . . . , α(Bk))
Ferguson has extended the idea of Bayesian estimation for a probability vector, where the conjugate prior would be the Dirchlet distribution, to the case where one wishes to estimate an entire probability measure

Summary

Introduction

In a nonparametric statistical model, the unknown of interest is the probability measure responsible for generating the data at hand. Polya tree distributions ([5], [6]) extend the idea of the Dirichlet process by defining the finite dimensional distributions as conditional probabilities in P. This allows for a specification of prior information which carries greater weight in subsets of where it is deemed appropriate according to the choice of a partition of. A point of contention is that the specification of the prior is partially dependent on the data Without this construction, the posterior from censored data will be a mixture of Polya trees. Our concluding section contains an example as well as a discussion on how the choice of a partition influences the posterior estimate

Polya Tree Distributions

Mixtures Of Polya Tree Distributions

An Application