A Bayesian method of sample size determination with practical applications

Summary.  The problem motivating the paper is the determination of sample size in clinical trials under normal likelihoods and at the substantive testing stage of a financial audit where normality is not an appropriate assumption. A combination of analytical and simulation-based techniques within the Bayesian framework is proposed. The framework accommodates two different prior distributions: one is the general purpose fitting prior distribution that is used in Bayesian analysis and the other is the expert subjective prior distribution, the sampling prior which is believed to generate the parameter values which in turn generate the data. We obtain many theoretical results and one key result is that typical non-informative prior distributions lead to very small sample sizes. In contrast, a very informative prior distribution may either lead to a very small or a very large sample size depending on the location of the centre of the prior distribution and the hypothesized value of the parameter. The methods that are developed are quite general and can be applied to other sample size determination problems. Some numerical illustrations which bring out many other aspects of the optimum sample size are given.     

Standardized posterior mode for the flexible use of a conjugate prior

The posterior mode under the standardized prior density is proposed to estimate a mean (vector) parameter, and its potential usefulness is discussed. Priors in this study include a conjugate prior and its generalized forms. When a prior density is factored into the standardized prior density and the supporting measure density, our suggestion is to discard the latter density and then to calculate the posterior mode of the mean under the standardized prior density. This treatment makes our choice of a prior density flexible. Implications of this treatment are discussed.     

Prbposterior expected loss as a scoring rule for prior distributions

A scoring rule for evaluating the usefulness of an assessed prior distribution should reflect the purpose for which the distribution is to be used. In this paper we suppose that sample data is to become available and that the posterior distribution will be used to estimate some quantity under a quadratic loss function. The utility of a prior distribution is consequently determined by its preposterior expected quadratic loss. It is shown that this loss function has properties desirable in a scoring rule and formulae are derived for calculating the scores it gives in some common problems. Many scoring rules give a very poor score to any improper prior distribution but, in contrast, the scoring rule proposed here provides a meaningful measure for comparing the usefulness of assessed prior distributions and non-informative (improper) prior distributions. Results for making this comparison in various situations are also given.     

On selecting the best treatment in a generalized linear model

The problem of selecting the best treatment is studied under generalized linear models. For certain balanced designs, it is shown that simple rules are Bayes with respect to any non-informative prior on the treatment effects under any monotone invariant loss. When the nuisance parameters such as block effects are assumed to follow a uniform (improper) prior or a normal prior, Bayes rules are obtained for the normal linear model under more suitable balanced designs, keeping the generality of the loss and the generality of the non-informativeness on the prior of the treatment effects. These results are extended to certain types of informative priors on the treatment effects. When the designs are unbalanced, algorithms based on the Gibbs sampler and the Laplace method are provided to compute the Bayes rules.     

Alternative posterior consistency results in nonparametric binary regression using Gaussian process priors

We establish consistency of posterior distribution when a Gaussian process prior is used as a prior distribution for the unknown binary regression function. Specifically, we take the work of Ghosal and Roy [2006. Posterior consistency of Gaussian process prior for nonparametric binary regression. Ann. Statist. 34, 2413–2429] as our starting point, and then weaken their assumptions on the smoothness of the Gaussian process kernel while retaining a stronger yet applicable condition about design points. Furthermore, we extend their results to multi-dimensional covariates under a weaker smoothness condition on the Gaussian process. Finally, we study the extent to which posterior consistency can be achieved under a general model where, when additional hyperparameters in the covariance function of a Gaussian process are involved.     

Bayes and robust Bayes predictions in a subfamily of scale parameters under a precautionary loss function

ABSTRACT

This paper deals with Bayes, robust Bayes, and minimax predictions in a subfamily of scale parameters under an asymmetric precautionary loss function. In Bayesian statistical inference, the goal is to obtain optimal rules under a specified loss function and an explicit prior distribution over the parameter space. However, in practice, we are not able to specify the prior totally or when a problem must be solved by two statisticians, they may agree on the choice of the prior but not the values of the hyperparameters. A common approach to the prior uncertainty in Bayesian analysis is to choose a class of prior distributions and compute some functional quantity. This is known as Robust Bayesian analysis which provides a way to consider the prior knowledge in terms of a class of priors Γ for global prevention against bad choices of hyperparameters. Under a scale invariant precautionary loss function, we deal with robust Bayes predictions of Y based on X. We carried out a simulation study and a real data analysis to illustrate the practical utility of the prediction procedure.     

Asymptotically Informative Prior for Bayesian Analysis

In classical Bayesian inference the prior is treated as fixed and its effects are ignored asymptotically, and useful information, if any, is wasted. However, in practice often an informative prior is summarized from previous similar or the same kind of studies, which contains useful cumulative information for the current study. We treat such prior to be non-fixed, i.e., we give the data sizes in the prior studies similar status as the that of the current dataset. Under this formulation, the prior is asymptotically non-negligible, and its original information is transferred to the new study. We explore some basic properties of Bayesian estimators under such prior formulation, and illustrate the method via simulation.     

Consistency of Bernstein polynomial posteriors

A Bernstein prior is a probability measure on the space of all the distribution functions on [0, 1]. Under very general assumptions, it selects absolutely continuous distribution functions, whose densities are mixtures of known beta densities. The Bernstein prior is of interest in Bayesian nonparametric inference with continuous data. We study the consistency of the posterior from a Bernstein prior. We first show that, under mild assumptions, the posterior is weakly consistent for any distribution function P ₀ on [0, 1] with continuous and bounded Lebesgue density. With slightly stronger assumptions on the prior, the posterior is also Hellinger consistent. This implies that the predictive density from a Bernstein prior, which is a Bayesian density estimate, converges in the Hellinger sense to the true density (assuming that it is continuous and bounded). We also study a sieve maximum likelihood version of the density estimator and show that it is also Hellinger consistent under weak assumptions. When the order of the Bernstein polynomial, i.e. the number of components in the beta distribution mixture, is truncated, we show that under mild restrictions the posterior concentrates on the set of pseudotrue densities. Finally, we study the behaviour of the predictive density numerically and we also study a hybrid Bayes–maximum likelihood density estimator.     

A comparison of the estimative and predictive methods of estimating posterior probabilities

An asymptotic account Is presented on the relative performance of the so-called estimative and predictive methods of estimating the posterior probability that an object belongs to one of two possible multivariate normal populations. For equal prior probabil-ities it is concluded that the predictive method generally gives a less extreme estimate than the estimative. This is supported by previously available results based essentially on simulation studies. Conditions under which the predictive method provides less extreme estimates for arbitrary prior probabilities are considered. Also, the asymptotic biases associated with the two methods are compared.     

Minimax Estimation of the Bounded Parameter of Some Discrete Distributions Under LINEX Loss Function

For a class of discrete distributions, including Poisson(θ), Generalized Poisson(θ), Borel(m, θ), etc., we consider minimax estimation of the parameter θ under the assumption it lies in a bounded interval of the form [0, m] and a LINEX loss function. Explicit conditions for the minimax estimator to be Bayes with respect to a boundary supported prior are given. Also for Bernoulli(θ)-distribution, which is not in the mentioned class of discrete distributions, we give conditions for which the Bayes estimator of θ ∈ [0, m], m < 1 with respect to a boundary supported prior is minimax under LINEX loss function. Numerical values are given for the largest values of m for which the corresponding Bayes estimators of θ are minimax.     

Calibrated Bayes factors for model comparison

A Bayes factor between two models can be greatly affected by the prior distributions on the model parameters. When prior information is weak, very dispersed proper prior distributions are known to create a problem for the Bayes factor when competing models differ in dimension, and it is of even greater concern when one of the models is of infinite dimension. Therefore, we propose an innovative method which uses training samples to calibrate the prior distributions so that they achieve a reasonable level of ‘information’. Then the calibrated Bayes factor can be computed over the remaining data. This method makes no assumption on model forms (parametric or nonparametric) and can be used with both proper and improper priors. We illustrate, through simulation studies and a real data example, that the calibrated Bayes factor yields robust and reliable model preferences under various situations.     

Application of Markov chain Monte Carlo methods to modelling birth prevalence of Down syndrome

Data collected before the routine application of prenatal screening are of unique value in estimating the natural live-birth prevalence of Down syndrome. However, much of these data are from births from over 20 years ago and they are of uncertain quality. In particular, they are subject to varying degrees of underascertainment. Published approaches have used ad hoc corrections to deal with this problem or have been restricted to data sets in which ascertainment is assumed to be complete. In this paper we adopt a Bayesian approach to modelling ascertainment and live-birth prevalence. We consider three prior specifications concerning ascertainment and compare predicted maternal-age-specific prevalence under these three different prior specifications. The computations are carried out by using Markov chain Monte Carlo methods in which model parameters and missing data are sampled.     

Bayes prediction based on right censored data

This paper is concerned with a Bayes prediction problem in the exponential distribution under random censorship. Using censored samples, we work out a prediction interval for a sum of interest which consists of some future samples. Differing from the general Bayes approach, we do not specify the prior distribution of the parameter, and only a first moment condition on the prior is assumed. Simulation studies are conducted to exhibit the coverage probabilities of the prediction interval. Financial support from the IAP research network (#P5/24) of the Belgian Government (Belgian Science Policy) is gratefully acknowledged.     

Reference priors for prediction

This article discusses the reference decision method for developing noninformative priors for prediction analyses. An information-theoretic criterion is advocated for choosing priors. Reference priors for prediction are defined to be priors which maximize the criterion in some asymptotic sense. These priors satisfy Jeffreys' original requirement of invariance under reparametrization. In the regular case, an explicit form of reference priors for prediction is given. Typically, it reduces to the Jeffreys prior. However, an example is given to illustrate how it produces a different prior than the ordinary noninformative priors.     

Bayesian nonparametric survival analysis for grouped data

A Bayesian nonparametric estimate of the survival distribution is derived under a particular sampling scheme for grouped data that includes the possibility of censoring. The estimate uses the prior information to smooth the data, giving an estimate which is continuous. As special cases survival estimates for life tables are obtained and the estimate of Susarla and Van Ryzin (1976) is derived. As the weight of the prior information tends to zero, the Bayesian estimate reduces to a continuous version of the nonparametric maximum-likelihood estimate. An empirical Bayes modification of the procedure is illustrated on a data set from Cutler and Ederer (1958).     

Bayesian inference for Poisson and multinomial log-linear models

Categorical data frequently arise in applications in the Social Sciences. In such applications, the class of log-linear models, based on either a Poisson or (product) multinomial response distribution, is a flexible model class for inference and prediction. In this paper we consider the Bayesian analysis of both Poisson and multinomial log-linear models. It is often convenient to model multinomial or product multinomial data as observations of independent Poisson variables. For multinomial data, Lindley (1964) [20] showed that this approach leads to valid Bayesian posterior inferences when the prior density for the Poisson cell means factorises in a particular way. We develop this result to provide a general framework for the analysis of multinomial or product multinomial data using a Poisson log-linear model. Valid finite population inferences are also available, which can be particularly important in modelling social data. We then focus particular attention on multivariate normal prior distributions for the log-linear model parameters. Here, an improper prior distribution for certain Poisson model parameters is required for valid multinomial analysis, and we derive conditions under which the resulting posterior distribution is proper. We also consider the construction of prior distributions across models, and for model parameters, when uncertainty exists about the appropriate form of the model. We present classes of Poisson and multinomial models, invariant under certain natural groups of permutations of the cells. We demonstrate that, if prior belief concerning the model parameters is also invariant, as is the case in a ‘reference’ analysis, then the choice of prior distribution is considerably restricted. The analysis of multivariate categorical data in the form of a contingency table is considered in detail. We illustrate the methods with two examples.     

On Maximum Depth and Related Classifiers

Abstract.  Over the last couple of decades, data depth has emerged as a powerful exploratory and inferential tool for multivariate data analysis with wide-spread applications. This paper investigates the possible use of different notions of data depth in non-parametric discriminant analysis. First, we consider the situation where the prior probabilities of the competing populations are all equal and investigate classifiers that assign an observation to the population with respect to which it has the maximum location depth. We propose a different depth-based classification technique for unequal prior problems, which is also useful for equal prior cases, especially when the populations have different scatters and shapes. We use some simulated data sets as well as some benchmark real examples to evaluate the performance of these depth-based classifiers. Large sample behaviour of the misclassification rates of these depth-based non-parametric classifiers have been derived under appropriate regularity conditions.     

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.     

Empirical bayes shrinkage estimation of reliability in the exponential distribution

In this paper we propose two empirical Bayes shrinkage estimators for the reliability of the exponential distribution and study their properties. Under the uniform prior distribution and the inverted gamma prior distribution these estimators are developed and compared with a preliminary test estimator and with a shrinkage testimator in terms of mean squared error. The proposed empirical Bayes shrinkage estimator under the inverted gamma prior distribution is shown to be preferable to the preliminary test estimator and the shrinkage testimator when the prior value of mean life is clsoe to the true mean life.     

Exponential-Bound Property of Estimators and Variable Selection in Generalized Additive Models

In this article, utilizing a scale mixture of skew-normal distribution in which mixing random variable is assumed to follow a mixture model with varying weights for each observation, we introduce a generalization of skew-normal linear regression model with the aim to provide resistant results. This model, which also includes the skew-slash distribution in a particular case, allows us to accommodate and detect outlying observations under the skew-normal linear regression model. Inferences about the model are carried out through the empirical Bayes approach. The conditions for propriety of the posterior and for existence of posterior moments are given under the standard noninformative priors for regression and scale parameters as well as proper prior for skewness parameter. Then, for Bayesian inference, a Markov chain Monte Carlo method is described. Since posterior results depend on the prior hyperparameters, we estimate them adopting the empirical Bayes method as well as using a Monte Carlo EM algorithm. Furthermore, to identify possible outliers, we also apply the Bayes factor obtained through the generalized Savage-Dickey density ratio. Examining the proposed approach on simulated instance and real data, it is found to provide not only satisfactory parameter estimates rather allow identifying outliers favorably.