On a Class of Random Probability Measures with General Predictive Structure

Abstract. In this study, we investigate a recently introduced class of non‐parametric priors, termed generalized Dirichlet process priors. Such priors induce (exchangeable random) partitions that are characterized by a more elaborate clustering structure than those arising from other widely used priors. A natural area of application of these random probability measures is represented by species sampling problems and, in particular, prediction problems in genomics. To this end, we study both the distribution of the number of distinct species present in a sample and the distribution of the number of new species conditionally on an observed sample. We also provide the Bayesian Non‐parametric estimator for the number of new species in an additional sample of given size and for the discovery probability as function of the size of the additional sample. Finally, the study of its conditional structure is completed by the determination of the posterior distribution.     

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.     

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 ratemaking under Dirichlet process mixtures

Experience ratemaking plays a crucial role in general insurance in determining future premiums of individuals in a portfolio by assessing observed claims from the whole portfolio. This paper investigates this problem in which claims can be modeled by certain parametric family of distributions. The Dirichlet process mixtures are employed to model the distributions of the parameters so as to make two advantages: to produce exact Bayesian experience premiums for a class of premium principles generated from generic error functions and, at the same time, provide robust and flexible ways to avoid possible bias caused by traditionally used priors such as non informative priors or conjugate priors. In this paper, the Bayesian experience ratemaking under Dirichlet process mixture models are investigated and due to the lack of analytical forms of the conditional expectations of the quantities concerned, the Gibbs sampling schemes are designed for the purpose of approximations.     

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.     

Predicting Home Run Production in Major League Baseball Using a Bayesian Semiparametric Model

This article attempts to predict home run hitting performance of Major League Baseball players using a Bayesian semiparametric model. Following Berry, Reese and Larkey we include in the model effects for era of birth, season of play, and home ball park. We estimate performance curves for each player using orthonormal quartic polynomials. We use a Dirichlet process prior on the unknown distribution for the coefficients of the polynomials, and parametric priors for the other effects. Dirichlet process priors are useful in prediction for two reasons: (1) an increased probability of obtaining more precise prediction comes with the increased flexibility of the prior specification, and (2) the clustering inherent in the Dirichlet process provides the means to share information across players. Data from 1871 to 2008 were used to fit the model. Data from 2009 to 2016 were used to test the predictive ability of the model. A parametric model was also fit to compare the predictive performance of the models. We used what we called “pure performance” curves to predict future performance for 22 players. The nonparametric method provided superior predictive performance.     

Natural (Non‐)Informative Priors for Skew‐symmetric Distributions

In this paper, we present an innovative method for constructing proper priors for the skewness (shape) parameter in the skew‐symmetric family of distributions. The proposed method is based on assigning a prior distribution on the perturbation effect of the shape parameter, which is quantified in terms of the total variation distance. We discuss strategies to translate prior beliefs about the asymmetry of the data into an informative prior distribution of this class. We show via a Monte Carlo simulation study that our non‐informative priors induce posterior distributions with good frequentist properties, similar to those of the Jeffreys prior. Our informative priors yield better results than their competitors from the literature. We also propose a scale‐invariant and location‐invariant prior structure for models with unknown location and scale parameters and provide sufficient conditions for the propriety of the corresponding posterior distribution. Illustrative examples are presented using simulated and real data.     

Bayesian spectral analysis models for quantile regression with Dirichlet process mixtures

This paper presents a Bayesian analysis of partially linear additive models for quantile regression. We develop a semiparametric Bayesian approach to quantile regression models using a spectral representation of the nonparametric regression functions and the Dirichlet process (DP) mixture for error distribution. We also consider Bayesian variable selection procedures for both parametric and nonparametric components in a partially linear additive model structure based on the Bayesian shrinkage priors via a stochastic search algorithm. Based on the proposed Bayesian semiparametric additive quantile regression model referred to as BSAQ, the Bayesian inference is considered for estimation and model selection. For the posterior computation, we design a simple and efficient Gibbs sampler based on a location-scale mixture of exponential and normal distributions for an asymmetric Laplace distribution, which facilitates the commonly used collapsed Gibbs sampling algorithms for the DP mixture models. Additionally, we discuss the asymptotic property of the sempiparametric quantile regression model in terms of consistency of posterior distribution. Simulation studies and real data application examples illustrate the proposed method and compare it with Bayesian quantile regression methods in the literature.     

Posterior Analysis for Normalized Random Measures with Independent Increments

Abstract.  One of the main research areas in Bayesian Nonparametrics is the proposal and study of priors which generalize the Dirichlet process. In this paper, we provide a comprehensive Bayesian non-parametric analysis of random probabilities which are obtained by normalizing random measures with independent increments (NRMI). Special cases of these priors have already shown to be useful for statistical applications such as mixture models and species sampling problems. However, in order to fully exploit these priors, the derivation of the posterior distribution of NRMIs is crucial: here we achieve this goal and, indeed, provide explicit and tractable expressions suitable for practical implementation. The posterior distribution of an NRMI turns out to be a mixture with respect to the distribution of a specific latent variable. The analysis is completed by the derivation of the corresponding predictive distributions and by a thorough investigation of the marginal structure. These results allow to derive a generalized Blackwell–MacQueen sampling scheme, which is then adapted to cover also mixture models driven by general NRMIs.     

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.     

Nonparametric Bayesian survival analysis using mixtures of Weibull distributions

Bayesian nonparametric methods have been applied to survival analysis problems since the emergence of the area of Bayesian nonparametrics. However, the use of the flexible class of Dirichlet process mixture models has been rather limited in this context. This is, arguably, to a large extent, due to the standard way of fitting such models that precludes full posterior inference for many functionals of interest in survival analysis applications. To overcome this difficulty, we provide a computational approach to obtain the posterior distribution of general functionals of a Dirichlet process mixture. We model the survival distribution employing a flexible Dirichlet process mixture, with a Weibull kernel, that yields rich inference for several important functionals. In the process, a method for hazard function estimation emerges. Methods for simulation-based model fitting, in the presence of censoring, and for prior specification are provided. We illustrate the modeling approach with simulated and real data.     

Incomplete categorical data analysis: a bayesian perspective

In this paper the Bayesian analysis of incomplete categorical data under informative general censoring proposed by Paulino and Pereira (1995) is revisited. That analysis is based on Dirichlet priors and can be applied to any missing data pattern. However, the known properties of the posterior distributions are scarce and therefore severe limitations to the posterior computations remain. Here is shown how a Monte Carlo simulation approach based on an alternative parameterisation can be used to overcome the former computational difficulties. The proposed simulation approach makes available the approximate estimation of general parametric functions and can be implemented in a very straightforward way.     

Non‐parametric Bayesian Intensity Model: Exploring Time‐to‐Event Data on Two Time Scales

Time‐to‐event data have been extensively studied in many areas. Although multiple time scales are often observed, commonly used methods are based on a single time scale. Analysing time‐to‐event data on two time scales can offer a more extensive insight into the phenomenon. We introduce a non‐parametric Bayesian intensity model to analyse two‐dimensional point process on Lexis diagrams. After a simple discretization of the two‐dimensional process, we model the intensity by a one‐dimensional piecewise constant hazard functions parametrized by the change points and corresponding hazard levels. Our prior distribution incorporates a built‐in smoothing feature in two dimensions. We implement posterior simulation using the reversible jump Metropolis–Hastings algorithm and demonstrate the applicability of the method using both simulated and empirical survival data. Our approach outperforms commonly applied models by borrowing strength in two dimensions.     

A Nonparametric Frailty Model for Clustered Survival Data

Clayton-type counting process formulations for survival data and parametric gamma models for cluster-specific frailty quantities are now routinely applied in analyses of clustered survival data. On the other hand, although nonparametric frailty models have been studied, they are not used much in practice. In this article, the distribution of the frailty terms is assumed to be an unknown random variable. The unknown frailty distribution is then modelled completely with a Dirichlet process prior. This prior assigns cluster units into sub-classes whose members have the same random frailty effect. The Gibbs sampler algorithm is used for computing posterior parameter estimates of the fixed effect hazards regression and the frailty distribution. The methodology is used to analyze community-clustered child survival in sub-Saharan Africa. The results show that the communities could be separated into fewer distinct classes of risk of childhood mortality; the fewer classes could be studied easily in order to provide useful guidance on the more effective use of resources for child health intervention programmes.     

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 semiparametric modeling and inference with mixtures of?symmetric distributions

We propose a semiparametric modeling approach for mixtures of symmetric distributions. The mixture model is built from a common symmetric density with different components arising through different location parameters. This structure ensures identifiability for mixture components, which is a key feature of the model as it allows applications to settings where primary interest is inference for the subpopulations comprising the mixture. We focus on the two-component mixture setting and develop a Bayesian model using parametric priors for the location parameters and for the mixture proportion, and a nonparametric prior probability model, based on Dirichlet process mixtures, for the random symmetric density. We present an approach to inference using Markov chain Monte Carlo posterior simulation. The performance of the model is studied with a simulation experiment and through analysis of a rainfall precipitation data set as well as with data on eruptions of the Old Faithful geyser.     

On priors that match posterior and frequentist distribution functions

In the multiparameter case, this paper characterizes priors so as to match, up to o(n^-1/2), the posterior joint cumulative distribution function (c.d.f.) of a posterior standardized version of the parametric vector with the corresponding frequentist c.d.f.     

A family of power‐divergence diagnostics for goodness‐of‐fit

The author extends to the Bayesian nonparametric context the multinomial goodness‐of‐fit tests due to Cressie & Read (1984). Her approach is suitable when the model of interest is a discrete distribution. She provides an explicit form for the tests, which are based on power‐divergence measures between a prior Dirichlet process that is highly concentrated around the model of interest and the corresponding posterior Dirichlet process. In addition to providing interesting special cases and useful approximations, she discusses calibration and the choice of test through examples.     

Two-sided Bayesian and frequentist tolerance intervals: general asymptotic results with applications

It is well known that that the construction of two-sided tolerance intervals is far more challenging than that of their one-sided counterparts. In a general framework of parametric models, we derive asymptotic results leading to explicit formulae for two-sided Bayesian and frequentist tolerance intervals. In the process, probability matching priors for such intervals are characterized and their role in finding frequentist tolerance intervals via a Bayesian route is indicated. Furthermore, in situations where matching priors are hard to obtain, we develop purely frequentist tolerance intervals as well. The findings are applied to real data. Simulation studies are seen to lend support to the asymptotic results in finite samples.     

Objective Bayesian inference for the ratio of the scale parameters of two Weibull distributions

The Weibull distribution is widely used due to its versatility and relative simplicity. In our paper, the non informative priors for the ratio of the scale parameters of two Weibull models are provided. The asymptotic matching of coverage probabilities of Bayesian credible intervals is considered, with the corresponding frequentist coverage probabilities. We developed the various priors for the ratio of two scale parameters using the following matching criteria: quantile matching, matching of distribution function, highest posterior density matching, and inversion of test statistics. One particular prior, which meets all the matching criteria, is found. Next, we derive the reference priors for groups of ordering. We see that all the reference priors satisfy a first-order matching criterion and that the one-at-a-time reference prior is a second-order matching prior. A simulation study is performed and an example given.