Rubbery Polya Tree

Abstract. Polya trees (PT) are random probability measures which can assign probability 1 to the set of continuous distributions for certain specifications of the hyperparameters. This feature distinguishes the PT from the popular Dirichlet process (DP) model which assigns probability 1 to the set of discrete distributions. However, the PT is not nearly as widely used as the DP prior. Probably the main reason is an awkward dependence of posterior inference on the choice of the partitioning subsets in the definition of the PT. We propose a generalization of the PT prior that mitigates this undesirable dependence on the partition structure, by allowing the branching probabilities to be dependent within the same level. The proposed new process is not a PT anymore. However, it is still a tail‐free process and many of the prior properties remain the same as those for the PT.     

Semiparametric Bayesian classification with longitudinal markers

Summary.  We analyse data from a study involving 173 pregnant women. The data are observed values of the β human chorionic gonadotropin hormone measured during the first 80 days of gestational age, including from one up to six longitudinal responses for each woman. The main objective in this study is to predict normal versus abnormal pregnancy outcomes from data that are available at the early stages of pregnancy. We achieve the desired classification with a semiparametric hierarchical model. Specifically, we consider a Dirichlet process mixture prior for the distribution of the random effects in each group. The unknown random-effects distributions are allowed to vary across groups but are made dependent by using a design vector to select different features of a single underlying random probability measure. The resulting model is an extension of the dependent Dirichlet process model, with an additional probability model for group classification. The model is shown to perform better than an alternative model which is based on independent Dirichlet processes for the groups. Relevant posterior distributions are summarized by using Markov chain Monte Carlo methods.     

Bayes and empirical bayes estimation of survival function under progressive censoring

The purpose of this note is to derive the Bayes and the empirical Bayes estimators of an unknown survival function F under progressively censored data with respect to the squared error loss function and a Dirichlet process prior using the fact that the posterior distribution of F given the data is a mixture of Dirichlet processes, and the assumption that the survival and the censor in0- distributions are continuous.     

Bayesian nonparametric inference for unimodal skew-symmetric distributions

This paper studies the case where the observations come from a unimodal and skew density function with an unknown mode. The skew-symmetric representation of such a density has a symmetric component which can be written as a scale mixture of uniform densities. A Dirichlet process (DP) prior is assigned to mixing distribution. We also assume prior distributions for the mode and the skewed component. A computational approach is used to obtain the Bayes estimate of the components. An example is given to illustrate the approach.     

Bayes Risk Consistency of Nonparametric Bayes Density Estimates2

Even though the literature on nonparametric density estimation is large, the literature on Bayesian estimation of the density function is relatively small. The reason is the lack of a suitable prior over the space of probability density functions. There have been attempts to define priors over the space of probability measures, but they have not yielded any workable prior for the purpose of density estimation. Dubins & Freedman (1963) have denned random distribution functions which are singular with probability one. Kraft (1964) has denned a class of distribution functions which have derivatives but not continuous derivatives and hence are not suitable for density estimation. The only really convenient prior is the Dirichlet process prior due to Ferguson (1973), but unfortunately this prior concentrates all its mass over the discrete distribution with a dense set of jumps. Recently Lo (1978) has overcome this difficulty by taking convolution of the Dirichlet process with a fixed continuous kernel. In Section 2, the existence of a version of the posterior distribution and the conditional expectation for arbitrary prior over the space of continuous density functions are discussed. The Bayes risk consistency of the Bayes estimator is discussed in Section 3. The Bayes estimator and its properties with respect to two specific prior distributions are discussed in Section 4. In Section 5 some negative results are presented. Finally a numerical example is given in Section 6.     

On several estimates to the precision parameter of Dirichlet process prior

The article presents careful comparisons among several empirical Bayes estimates to the precision parameter of Dirichlet process prior, with the setup of univariate observations and multigroup data. Specifically, the data are equipped with a two-stage compound sampling model, where the prior is assumed as a Dirichlet process that follows within a Bayesian nonparametric framework. The precision parameter α measures the strength of the prior belief and kinds of estimates are generated on the basis of observations, including the naive estimate, two calibrated naive estimates, and two different types of maximum likelihood estimates stemming from distinct distributions. We explore some theoretical properties and provide explicitly detailed comparisons among these estimates, in the perspectives of bias, variance, and mean squared error. Besides, we further present the corresponding calculation algorithms and numerical simulations to illustrate our theoretical achievements.     

A Method to Handle Zero Counts in the Multinomial Model

In the context of an objective Bayesian approach to the multinomial model, Dirichlet(a, …, a) priors with a < 1 have previously been shown to be inadequate in the presence of zero counts, suggesting that the uniform prior (a = 1) is the preferred candidate. In the presence of many zero counts, however, this prior may not be satisfactory either. A model selection approach is proposed, allowing for the possibility of zero parameters corresponding to zero count categories. This approach results in a posterior mixture of Dirichlet distributions and marginal mixtures of beta distributions, which seem to avoid the problems that potentially result from the various proposed Dirichlet priors, in particular in the context of extreme data with zero counts.     

Flexible clustering via hidden hierarchical Dirichlet priors

The Bayesian approach to inference stands out for naturally allowing borrowing information across heterogeneous populations, with different samples possibly sharing the same distribution. A popular Bayesian nonparametric model for clustering probability distributions is the nested Dirichlet process, which however has the drawback of grouping distributions in a single cluster when ties are observed across samples. With the goal of achieving a flexible and effective clustering method for both samples and observations, we investigate a nonparametric prior that arises as the composition of two different discrete random structures and derive a closed-form expression for the induced distribution of the random partition, the fundamental tool regulating the clustering behavior of the model. On the one hand, this allows to gain a deeper insight into the theoretical properties of the model and, on the other hand, it yields an MCMC algorithm for evaluating Bayesian inferences of interest. Moreover, we single out limitations of this algorithm when working with more than two populations and, consequently, devise an alternative more efficient sampling scheme, which as a by-product, allows testing homogeneity between different populations. Finally, we perform a comparison with the nested Dirichlet process and provide illustrative examples of both synthetic and real data.     

Reanalyzing Ultimatum Bargaining—Comparing Nondecreasing Curves Without Shape Constraints

We employ a hierarchical Bayesian method with exchangeable prior distributions to estimate and compare similar nondecreasing response curves. A Dirichlet process distribution is assigned to each of the response curves as a first stage prior. A second stage prior is then used to model the hyperparameters. We define parameters which will be used to compare the response curves. A Markov chain Monte Carlo method is applied to compute the resulting Bayesian estimates. To illustrate the methodology, we re-examine data from an experiment designed to test whether experimenter observation influences the ultimatum game. A major restriction of the original analysis was the shape constraint that the present technique allows us to greatly relax. We also consider independent priors and use Bayes factors to compare various models.     

On the Bayesian analysis of categorical data: the problem of nonresponse

It is demonstrated how a suitably chosen prior for the frequency parameters can streamline the Bayesian analysis of categorical data with missing entries due to nonresponse or other causes. The two cases where the data follow the Multinomial or the Hypergeometric model are treated separately. In the first case it is adequate to restrict the prior (for the cell probabilities) to the class of Dirichlet distributions. In the case of the Hypergeometric model it is convenient to select a prior from the class of Dirichlet-Multinomial (DM) distributions. The DM distributions are studied in some details.     

A note on the finite-dimensional Dirichlet prior

As an approximation to the Dirichlet process which involves the infinite-dimensional distribution, finite-dimensional Dirichlet prior is a widely appreciated method to model the underlying distribution in non parametric Bayesian analysis. In this short note, we present some key characteristics of finite-dimensional Dirichlet process and exploit some important sampling properties which are very useful in Bayesian non parametric/semiparametric analysis.     

Bayesian semiparametric modeling for stochastic precedence,with applications in epidemiology and survival analysis

We propose a prior probability model for two distributions that are ordered according to a stochastic precedence constraint, a weaker restriction than the more commonly utilized stochastic order constraint. The modeling approach is based on structured Dirichlet process mixtures of normal distributions. Full inference for functionals of the stochastic precedence constrained mixture distributions is obtained through a Markov chain Monte Carlo posterior simulation method. A motivating application involves study of the discriminatory ability of continuous diagnostic tests in epidemiologic research. Here, stochastic precedence provides a natural restriction for the distributions of test scores corresponding to the non-infected and infected groups. Inference under the model is illustrated with data from a diagnostic test for Johne’s disease in dairy cattle. We also apply the methodology to the comparison of survival distributions associated with two distinct conditions, and illustrate with analysis of data on survival time after bone marrow transplantation for treatment of leukemia.     

A mixture of beta–Dirichlet processes prior for Bayesian analysis of event history data

In this paper, we propose a mixture of beta–Dirichlet processes as a nonparametric prior for the cumulative intensity functions of a Markov process. This family of priors is a natural extension of a mixture of Dirichlet processes or a mixture of beta processes which are devised to compromise advantages of parametric and nonparametric approaches. They give most of their prior mass to the small neighborhood of a specific parametric model. We show that a mixture of beta–Dirichlet processes prior is conjugate with Markov processes. Formulas for computing the posterior distribution are derived. Finally, results of analyzing credit history data are given.     

Topics on Dynamic Panel Data Models with Random Effects Using Semi-Parametric Bayesian Approach

A common assumption in fitting panel data models is normality of stochastic subject effects. This can be extremely restrictive, making vague most potential features of true distributions. The objective of this article is to propose a modeling strategy, from a semi-parametric Bayesian perspective, to specify a flexible distribution for the random effects in dynamic panel data models. This is addressed here by assuming the Dirichlet process mixture model to introduce Dirichlet process prior for the random-effects distribution. We address the role of initial conditions in dynamic processes, emphasizing on joint modeling of start-up and subsequent responses. We adopt Gibbs sampling techniques to approximate posterior estimates. These important topics are illustrated by a simulation study and also by testing hypothetical models in two empirical contexts drawn from economic studies. We use modified versions of information criteria to compare the fitted models.     

A comparison of nonparametric priors in hierarchical mixture modelling for AFT regression

We will pursue a Bayesian nonparametric approach in the hierarchical mixture modelling of lifetime data in two situations: density estimation, when the distribution is a mixture of parametric densities with a nonparametric mixing measure, and accelerated failure time (AFT) regression modelling, when the same type of mixture is used for the distribution of the error term. The Dirichlet process is a popular choice for the mixing measure, yielding a Dirichlet process mixture model for the error; as an alternative, we also allow the mixing measure to be equal to a normalized inverse-Gaussian prior, built from normalized inverse-Gaussian finite dimensional distributions, as recently proposed in the literature. Markov chain Monte Carlo techniques will be used to estimate the predictive distribution of the survival time, along with the posterior distribution of the regression parameters. A comparison between the two models will be carried out on the grounds of their predictive power and their ability to identify the number of components in a given mixture density.     

Bayesian Semiparametric Modelling in Quantile Regression

Abstract. We propose a Bayesian semiparametric methodology for quantile regression modelling. In particular, working with parametric quantile regression functions, we develop Dirichlet process mixture models for the error distribution in an additive quantile regression formulation. The proposed non‐parametric prior probability models allow the shape of the error density to adapt to the data and thus provide more reliable predictive inference than models based on parametric error distributions. We consider extensions to quantile regression for data sets that include censored observations. Moreover, we employ dependent Dirichlet processes to develop quantile regression models that allow the error distribution to change non‐parametrically with the covariates. Posterior inference is implemented using Markov chain Monte Carlo methods. We assess and compare the performance of our models using both simulated and real data sets.     

Bayesian clustering and product partition models

Summary. We present a decision theoretic formulation of product partition models (PPMs) that allows a formal treatment of different decision problems such as estimation or hypothesis testing and clustering methods simultaneously. A key observation in our construction is the fact that PPMs can be formulated in the context of model selection. The underlying partition structure in these models is closely related to that arising in connection with Dirichlet processes. This allows a straightforward adaptation of some computational strategies—originally devised for nonparametric Bayesian problems—to our framework. The resulting algorithms are more flexible than other competing alternatives that are used for problems involving PPMs. We propose an algorithm that yields Bayes estimates of the quantities of interest and the groups of experimental units. We explore the application of our methods to the detection of outliers in normal and Student t regression models, with clustering structure equivalent to that induced by a Dirichlet process prior. We also discuss the sensitivity of the results considering different prior distributions for the partitions.     

Exact and approximate sum representations for the Dirichlet process

The Dirichlet process can be regarded as a random probability measure for which the authors examine various sum representations. They consider in particular the gamma process construction of Ferguson (1973) and the “stick‐breaking” construction of Sethuraman (1994). They propose a Dirichlet finite sum representation that strongly approximates the Dirichlet process. They assess the accuracy of this approximation and characterize the posterior that this new prior leads to in the context of Bayesian nonpara‐metric hierarchical models.     

Subjective priors for the dirichlet process

This paper reviews difficulties with the interpretation and use of the prior parameter u required in the Dirichlet approach to nonpararnetric Bayesian statistics. Two subjective prior distributions are introduced and studied. These priors are obtained computationally by requiring that the experimenter specify certain constraints.     

Nonparametric Bayesian estimation of survival function under random left truncation

Several observational studies give rise to randomly left truncated data. In a nonparametric model for such data X denotes a variable of interest, T denotes the truncation variable and the distributions of both X and T are left unspecified. For this model, the product-limit estimator, which is also the maximum likelihood estimator of the survival curve, has been widely discussed. In this article, a nonparametric Bayes estimator of the survival function based on randomly left truncated data and Dirichlet process prior is presented. Some new results on the mixtures of Dirichlet processes in the context of truncated data are obtained. These results are then used to derive the Bayes estimator of the survival function under squared error loss. The weak convergence of the Bayes estimator is studied. An example using transfusion related AIDS data quoted in Kalbfleisch and Lawless (1989) is considered.