The definition and elementary properties of Polya tree distributions are reviewed. Two theorems are presented showing that Polya trees can be constructed to concentrate arbitrarily closely about any desired pdf, and that Polya tree priors can put positive mass in every relative entropy neighborhood of every positive density with finite entropy, thereby satisfying a consistency condition. Such theorems are false for Dirichlet processes. Models are constructed combining partially specified Polya trees with other information such as monotonicity or unimodality. It is shown how to compute bounds on posterior expectations over the class of all priors with the given specifications. A numerical example is given. A theorem of Diaconis and Freedman about Dirichlet processes is generalized to Polya trees, allowing Polya trees to be the models for errors in regression problems. Finally, empirical Bayes models using Dirichlet processes are generalized to Polya trees. An example from Berry and Christensen is reanalyzed with a Polya tree model.

/pdf/some-aspects-of-polya-tree-distributions-for-statistical-4bsblgykri.pdf

Some Aspects of Polya Tree Distributions for Statistical Modelling

There are two generations of Gibbs sampling methods for semiparametric models involving the Dirichlet process. The first generation suffered from a severe drawback: the locations of the clusters, or groups of parameters, could essentially become fixed, moving only rarely. Two strategies that have been proposed to create the second generation of Gibbs samplers are integration and appending a second stage to the Gibbs sampler wherein the cluster locations are moved. We show that these same strategies are easily implemented for the sequential importance sampler, and that the first strategy dramatically improves results. As in the case of Gibbs sampling, these strategies are applicable to a much wider class of models. They are shown to provide more uniform importance sampling weights and lead to additional Rao-Blackwellization of estimators.



Deux generations d'echantillonneurs de Gibbs ont ete popularisees pour les modeles semi-parametriques ou intervient le processus de Dirichlet. La premiere generation etait lourdement handicapee du fait que la localisation des grappes, ou groupes de parametres, pouvait devenir quasi-stationnaire. La deuxieme generation d'echantillonneurs a comble cette lacune en incorporant deux strategies: I'integration et I'ajout d'une eeape de relocalisation des grappes. Les auteurs montrent ici que ces strategies sont egalement applicables a la methode d'echantillonnage sequentiel pondere et que I'integration est particulierement efficace dans ce contexte. Comme dans le cas de l'echantillonnage de Gibbs, les deux strategies s'appliquent en fait a une classe beaucoup plus grande de modeles. Comme les auteurs le font valoir, elles produisent des poids d'echantillonnage plus uniformes et permettent une Rao-Blackwellisation additionnelle des estimateurs.

Sequential importance sampling for nonparametric Bayes models: The next generation

This article surveys Bayesian methods for categorical data analysis, with primary emphasis on contingency table analysis. Early innovations were proposed by Good (1953, 1956, 1965) for smoothing proportions in contingency tables and by Lindley (1964) for inference about odds ratios. These approaches primarily used conjugate beta and Dirichlet priors. Altham (1969, 1971) presented Bayesian analogs of small-sample frequentist tests for 2 x 2 tables using such priors. An alternative approach using normal priors for logits received considerable attention in the 1970s by Leonard and others (e.g., Leonard 1972). Adopted usually in a hierarchical form, the logit-normal approach allows greater flexibility and scope for generalization. The 1970s also saw considerable interest in loglinear modeling. The advent of modern computational methods since the mid-1980s has led to a growing literature on fully Bayesian analyses with models for categorical data, with main emphasis on generalized linear models such as logistic regression for binary and multi-category response variables.

http://udsmed.u-strasbg.fr/labiostat/IMG/pdf/AGRESTI_bayesian_inference_for_categorical_data_analysis.pdf

Bayesian inference for categorical data analysis

The computation of Bayes estimators based on mixtures of Dirichlet processes is treated in this article. These estimators may be written as ratios of two multidimensional integrals, each of which may be decomposed into a weighted average of products of one-dimensional integrals. An importance sampling Monte Carlo method is proposed to approximate each of the weighted averages. A prior error bound for each of the Monte Carlo estimators and a posterior error bound for the ratio are developed to measure the efficiency of the Monte Carlo method. Jackknife and random group error estimates are also considered. Two examples are given which illustrate the computation of the Bayes estimators.

Computations of mixtures of dirichlet processes

Reliability analysis : Optimal inspection and maintenance schedules of failing systems

SUMMARY A smooth empirical Bayes estimator for a binomial parameter is derived. The risk in using this estimator is compared by simulation with that of eight other binomial estimators. For those estimators not having a simple closed form risk, suitable regression equations on the simulation parameters are obtained. These equations are used to identify the regions of superiority of each of the estimators. A numerical example is provided.

https://www.jstor.org/stable/info/2334734

Empirical Bayes estimation of the binomial parameter

Maritz (1966) and Lemon & Krutchkoff (1969) each describe discrete empirical Bayes smoothing techniques. These techniques essentially attempt to approximate the prior distribution function. Here a continuous smoothing technique which is based on a smooth and continuous approximation to the prior density function is presented. Results from a Monte Carlo study of the Poisson distribution are reported which show that the continuous smoothing technique has desirable small-sample properties. Some comparisons with discrete smoothing techniques are also made.

A continuous empirical Bayes smoothing technique

An empirical Bayes estimator is given for the scale parameter in the two-parameter Weibull distribution. The scale parameter is assumed to vary randomly throughout a sequence of experiments according to a common, but unknown, prior distribution. The shape parameter is assumed to be known, however, it may be different in each experiment. The estimator is obtained by means of a continuous approximation to the unknown prior density function. Results from Monte Carlo simulation are reported which show that the estimator has smaller mean-squared errors than the usual maximum-likelihood estimator.

H. F. Martz

Papers

Empirical Bayes estimation of the binomial parameter

A continuous empirical Bayes smoothing technique

An empirical bayes estimator for the scale parameter of the two‐parameter weibull distribution