Showing papers on "Hierarchical Dirichlet process published in 1997"

Beta-Stacy processes and a generalization of the Pólya-urn scheme

Abstract: beta-Stacy process is dened. It is shown to be neutral to the right and a generalization of the Dirichlet process. The posterior distribution is also a beta-Stacy process given independent and identically distributed (iid) observations, possibly with right censoring, from F. A generalization of the P olya-urn scheme is introduced which characterizes the discrete betaStacy process. 1. Introduction. Let F be the space of cumulative distribution functions (cdfs) oni0;1e. This paper considers placing a probability distribution on F by dening a stochastic process F onei0;1e; Ae, where A is the Borel eld of subsets, such that Fe0e D 0 a.s., F is a.s. nondecreasing, a.s. right continuous and lim t!1 Fete D 1 a.s. Thus, with probability 1, the sample paths of F are cdf’s. Previous work includes the Dirichlet process [Ferguson (1973, 1974)], neutral to the right processes [Doksum (1974)], the extended gamma process [Dykstra and Laud (1981)], the beta process [Hjort (1990)] and P olya trees [Lavine (1992, 1994), Mauldin, Sudderth and Williams (1992)]. The purpose of this paper is twofold: (1) to introduce a new stochastic process which generalizes the Dirichlet process, in that more exible prior beliefs are able to be represented, and, unlike the Dirichlet process, is conjugate to right censored observations, and (2) to introduce a generalization of the P olyaurn scheme in order to characterize the discrete time version of the process. The property of conjugacy to right censored observations is also a feature of the beta process; however, with the beta process the statistician is required to consider hazard rates and cumulative hazards when constructing the prior. The beta-Stacy process only requires considerations on the distribution of the observations. The process is shown to be neutral to the right. The present paper is restricted to considering the estimation of an unknown cdf oni0;1e, although it is trivially extended to includee1 ;1e. Finally, for ease of notation, F is written to mean either the cdf or the corresponding probability measure. The organization of the paper is as follows. In Section 2 the process is dened and its connections with other processes given. We also provide an

Learning and Updating of Uncertainty in Dirichlet Models

Abstract: In this paper we analyze the problem of learning and updating of uncertainty in Dirichlet models, where updating refers to determining the conditional distribution of a single variable when some evidence is known. We first obtain the most general family of prior-posterior distributions which is conjugate to a Dirichlet likelihood and we identify those hyperparameters that are influenced by data values. Next, we describe some methods to assess the prior hyperparameters and we give a numerical method to estimate the Dirichlet parameters in a Bayesian context, based on the posterior mode. We also give formulas for updating uncertainty by determining the conditional probabilities of single variables when the values of other variables are known. A time series approach is presented for dealing with the cases in which samples are not identically distributed, that is, the Dirichlet parameters change from sample to sample. This typically occurs when the population is observed at different times. Finally, two examples are given that illustrate the learning and updating processes and the time series approach.

A note on the scale parameter of the Dirichlet Process

Abstract: This paper gives an interpretation for the scale parameter of a Dirichlet process when the aim is to estimate a linear functional of an unknown probability distribution. We provide exact first and second posterior moments for such functionals under both informative and noninformative prior specifications. The noninformative case provides a normal approximation to the Bayesian bootstrap. RESUME