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.     

Eliciting expert judgements about a set of proportions

Eliciting expert knowledge about several uncertain quantities is a complex task when those quantities exhibit associations. A well-known example of such a problem is eliciting knowledge about a set of uncertain proportions which must sum to 1. The usual approach is to assume that the expert's knowledge can be adequately represented by a Dirichlet distribution, since this is by far the simplest multivariate distribution that is appropriate for such a set of proportions. It is also the most convenient, particularly when the expert's prior knowledge is to be combined with a multinomial sample since then the Dirichlet is the conjugate prior family. Several methods have been described in the literature for eliciting beliefs in the form of a Dirichlet distribution, which typically involve eliciting from the expert enough judgements to identify uniquely the Dirichlet hyperparameters. We describe here a new method which employs the device of over-fitting, i.e. eliciting more than the minimal number of judgements, in order to (a) produce a more carefully considered Dirichlet distribution and (b) ensure that the Dirichlet distribution is indeed a reasonable fit to the expert's knowledge. The method has been implemented in a software extension of the Sheffield elicitation framework (SHELF) to facilitate the multivariate elicitation process.     

Exact filtering for partially observed continuous time models

Summary.  The forward–backward algorithm is an exact filtering algorithm which can efficiently calculate likelihoods, and which can be used to simulate from posterior distributions. Using a simple result which relates gamma random variables with different rates, we show how the forward–backward algorithm can be used to calculate the distribution of a sum of gamma random variables, and to simulate from their joint distribution given their sum. One application is to calculating the density of the time of a specific event in a Markov process, as this time is the sum of exponentially distributed interevent times. This enables us to apply the forward–backward algorithm to a range of new problems. We demonstrate our method on three problems: calculating likelihoods and simulating allele frequencies under a non-neutral population genetic model, analysing a stochastic epidemic model and simulating speciation times in phylogenetics.     

On the Posterior Consistency of Mixtures of Dirichlet Process Priors with Censored Data

Mixtures of Dirichlet process priors offer a reasonable compromise between purely parametric and purely non‐parametric models, and are popularly used in survival analysis and for testing problems with non‐parametric alternatives. In this paper, we study large sample properties of the posterior distribution with a mixture of Dirichlet process priors. We show that the posterior distribution of the survival function is consistent with right censored data.     

On the Dirichlet distributions on symmetric matrices

In this paper, we introduce a generalization of the Dirichlet distribution on symmetric matrices which represents the multivariate version of the Connor and Mosimann generalized real Dirichlet distribution. We establish some properties concerning this generalized distribution. We also extend to the matrix Dirichlet distribution a remarkable characterization established in the real case by Darroch and Ratcliff.     

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.     

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 New Procedure to Monitor the Mean of a Quality Characteristic

The Shewhart, Bonferroni-adjustment, and analysis of means (ANOM) control charts are typically applied to monitor the mean of a quality characteristic. The Shewhart and Bonferroni procedure are utilized to recognize special causes in production process, where the control limits are constructed by assuming normal distribution for known parameters (mean and standard deviation), and approximately normal distribution regarding to unknown parameters. The ANOM method is an alternative to the analysis of variance method. It can be used to establish the mean control charts by applying equicorrelated multivariate non central t distribution. In this article, we establish new control charts, in phases I and II monitoring, based on normal and t distributions having as a cause a known (or unknown) parameter (standard deviation). Our proposed methods are at least as effective as the classical Shewhart methods and have some advantages.     

Goodness-of-fit tests based on the distance between the Dirichlet process and its base measure

The Dirichlet process is a fundamental tool in studying Bayesian nonparametric inference. The Dirichlet process has several sum representations, where each one of these representations highlights some aspects of this important process. In this paper, we use the sum representations of the Dirichlet process to derive explicit expressions that are used to calculate Kolmogorov, Lévy, and Cramér–von Mises distances between the Dirichlet process and its base measure. The derived expressions of the distance are used to select a proper value for the concentration parameter of the Dirichlet process. These tools are also used in a goodness-of-fit test. Illustrative examples and simulation results are included.     

Search With Learning for Differentiated Products: Evidence from E-Commerce

This article provides a method to estimate search costs in a differentiated product environment in which consumers are uncertain about the utility distribution. Consumers learn about the utility distribution by Bayesian updating their Dirichlet process prior beliefs. The model provides expressions for bounds on the search costs that can rationalize observed search and purchasing behavior. Using individual-specific data on web browsing and purchasing behavior for MP3 players sold online we show how to use these bounds to estimate search costs as well as the parameters of the utility distribution. Our estimates indicate that search costs are sizable. We show that ignoring consumer learning while searching can lead to severely biased search cost and elasticity estimates.     

Using the chi-square statistic to monitor compositional process data

SUMMARY We investigate the use of the chi-square control chart as a simple multivariate method for shopfloor monitoring of compositional process data. Although this chart is usually considered to be applicable only with multinomial process data, we show that it is also valid, in a certain asymptotic sense, for compositional data that arise from the Dirichlet distribution. For general compositional data, we show that the chi-square statistic can be used for process monitoring, provided that we make a simple adjustment to the degrees of freedom in the chi-square reference distribution. This method is illustrated and compared in four examples with the T 2 chart based on log-ratio transformation of the data.     

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.     

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 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 semiparametric pivotal bayesian inference for quantiles

In semiparametric inference we distinguish between the parameter of interest which may be a location parameter, and a nuisance parameter that determines the remaining shape of the sampling distribution. As was pointed out by Diaconis and Freedman the main problem in semiparametric Bayesian inference is to obtain a consistent posterior distribution for the parameter of interest. The present paper considers a semiparametric Bayesian method based on a pivotal likelihood function. It is shown that when the parameter of interest is the median, this method produces a consistent posterior distribution and is easily implemented, Numerical comparisons with classical methods and with Bayesian methods based on a Dirichlet prior are provided. It is also shown that in the case of symmetric intervals, the classical confidence coefficients have a Bayesian interpretation as the limiting posterior probability of the interval based on the Dirichlet prior with a parameter that converges to zero.     

Estimations of the parameter of a Dirichlet distribution using residual allocation model representations and sampling properties

Estimating the parameter of a Dirichlet distribution is an interesting question since this distribution arises in many situations of applied probability. Classical procedures are based on sample of Dirichlet distribution. In this paper we exhibit five different estimators from only one observation. They are based either on residual allocation model decompositions or on sampling properties of Dirichlet distributions. Two ways are investigated: the first one uses fragments’ size and the second one uses size-biased permutations of a partition. Numerical computations based on simulations are supplied. The estimators are finally used to estimate birth probabilities per month.     

A Semiparametric Bayesian Model for Circular-Linear Regression

Circular data are observations that are represented as points on a unit circle. Times of day and directions of wind are two such examples. In this work, we present a Bayesian approach to regress a circular variable on a linear predictor. The regression coefficients are assumed to have a nonparametric distribution with a Dirichlet process prior. The semiparametric Bayesian approach gives added flexibility to the model and is useful especially when the likelihood surface is ill behaved. Markov chain Monte Carlo techniques are used to fit the proposed model and to generate predictions. The method is illustrated using an environmental data set.     

A Bayesian semiparametric accelerated failure time model for arbitrarily censored data with covariates subject to measurement error

A flexible Bayesian semiparametric accelerated failure time (AFT) model is proposed for analyzing arbitrarily censored survival data with covariates subject to measurement error. Specifically, the baseline error distribution in the AFT model is nonparametrically modeled as a Dirichlet process mixture of normals. Classical measurement error models are imposed for covariates subject to measurement error. An efficient and easy-to-implement Gibbs sampler, based on the stick-breaking formulation of the Dirichlet process combined with the techniques of retrospective and slice sampling, is developed for the posterior calculation. An extensive simulation study is conducted to illustrate the advantages of our approach.     

Random Bernstein Polynomials

Random Bernstein polynomials which are also probability distribution functions on the closed unit interval are studied. The probability law of a Bernstein polynomial so defined provides a novel prior on the space of distribution functions on [0, 1] which has full support and can easily select absolutely continuous distribution functions with a continuous and smooth derivative. In particular, the Bernstein polynomial which approximates a Dirichlet process is studied. This may be of interest in Bayesian non-parametric inference. In the second part of the paper, we study the posterior from a "Bernstein–Dirichlet" prior and suggest a hybrid Monte Carlo approximation of it. The proposed algorithm has some aspects of novelty since the problem under examination has a "changing dimension" parameter space.     

Application of Dirichlet mixture of normals in growth curve models

In this paper, we present growth curve models with an auxiliary variable which contains an uncertain data distribution based on mixtures of standard components, such as normal distributions. The multimodality of the auxiliary random variable motivates and necessitates the use of mixtures of normal distributions in our model. We have observed that Dirichlet process priors, composed of discrete and continuous components, are appropriate in addressing the two problems of determining the number of components and estimating the parameters simultaneously and are especially useful in the aforementioned multimodal scenario. A model for the application of Dirichlet mixture of normals (DMN) in growth curve models under Bayesian formulation is presented and algorithms for computing the number of components, as well as estimating the parameters are also rendered. The simulation results show that our model gives improved goodness of fit statistics over models without DMN and the estimates for the number of components and for parameters are reasonably accurate.