Objective priors for hypothesis testing in one-way random effects models

The Bayes factor is a key tool in hypothesis testing. Nevertheless, the important issue of which priors should be used to develop objective Bayes factors remains open. The authors consider this problem in the context of the one-way random effects model. They use concepts such as orthogonality, predictive matching and invariance to justify a specific form of the priors for common parameters and derive the intrinsic and divergence based prior for the new parameter. The authors show that both intrinsic priors or divergence-based priors produce consistent Bayes factors. They illustrate the methods and compare them with other proposals.     

A Note on Priors for the Multinomial Model

An “overall objective” prior proposed for the multinomial model is shown to be inadequate in the presence of zero counts. An earlier proposed reference prior for when interest is in a particular category suffers from similar problems. It is argued that there is no need to deviate from the uniform prior proposed by Jeffreys, for which links with a non-Bayesian approach, when prediction is of interest, are shown.     

Adaptive power priors with empirical Bayes for clinical trials

Incorporating historical information into the design and analysis of a new clinical trial has been the subject of much discussion as a way to increase the feasibility of trials in situations where patients are difficult to recruit. The best method to include this data is not yet clear, especially in the case when few historical studies are available. This paper looks at the power prior technique afresh in a binomial setting and examines some previously unexamined properties, such as Box P values, bias, and coverage. Additionally, it proposes an empirical Bayes‐type approach to estimating the prior weight parameter by marginal likelihood. This estimate has advantages over previously criticised methods in that it varies commensurably with differences in the historical and current data and can choose weights near 1 when the data are similar enough. Fully Bayesian approaches are also considered. An analysis of the operating characteristics shows that the adaptive methods work well and that the various approaches have different strengths and weaknesses.     

Reference priors for exponential families

Reference analysis, introduced by Bernardo (J. Roy. Statist. Soc. 41 (1979) 113) and further developed by Berger and Bernardo (On the development of reference priors (with discussion). In: J.M. Bernardo, J.O. Berger, A.P. Dawid, A.F.M. Smith (Eds.), Bayesian Statistics, Vol. 4, Clarendon Press, Oxford, pp. 35–60), has proved to be one of the most successful general methods to derive noninformative prior distributions. In practice, however, reference priors are typically difficult to obtain. In this paper we show how to find reference priors for a wide class of exponential family likelihoods.     

Kullback-Leibler divergence to evaluate posterior sensitivity to different priors for autoregressive time series models

In this paper we use the Kullback-Leibler divergence to measure the distance between the posteriors of the autoregressive (AR) model coefficients, aiming to evaluate mathematically the sensitivity of the coefficients posterior to different types of priors, i.e. Jeffreys’, g, and natural conjugate priors. In addition, we evaluate the impact of the posteriors distance in Bayesian estimates of mean and variance of the model coefficients by generating a large number of Monte Carlo simulations from the posteriors. Simulation study results show that the coefficients posterior is sensitive to prior distributions, and the posteriors distance has more influence on Bayesian estimates of variance than those of mean of the model coefficients. Same results are obtained from the application to real-world time series datasets.     

Probability matching priors for an extended statistical calibration model

Statistical calibration or inverse prediction involves data collected in two stages. In the first stage, several values of an endogenous variable are observed, each corresponding to a known value of an exogenous variable; in the second stage, one or more values of the endogenous variable are observed which correspond to an unknown value of the exogenous variable. When estimating the value of the latter, it has been suggested that the variability about the regression relationship should not be assumed to be equal for the two stages of data collection. In this paper, the authors present a Bayesian method of analysis based on noninformative priors that takes this heteroscedasticity into account.     

Objective Bayesian analysis for a spatial model with nugget effects

The focus of this paper is objective priors for spatially correlated data with nugget effects. In addition to the Jeffreys priors and commonly used reference priors, two types of “exact” reference priors are derived based on improper marginal likelihoods. An “equivalence” theorem is developed in the sense that the expectation of any function of the score functions of the marginal likelihood function can be taken under marginal likelihoods. Interestingly, these two types of reference priors are identical.     

Non parametric Bayesian analysis of the two-sample problem with censoring

Testing for differences between two groups is a fundamental problem in statistics, and due to developments in Bayesian non parametrics and semiparametrics there has been renewed interest in approaches to this problem. Here we describe a new approach to developing such tests and introduce a class of such tests that take advantage of developments in Bayesian non parametric computing. This class of tests uses the connection between the Dirichlet process (DP) prior and the Wilcoxon rank sum test but extends this idea to the DP mixture prior. Here tests are developed that have appropriate frequentist sampling procedures for large samples but have the potential to outperform the usual frequentist tests. Extensions to interval and right censoring are considered and an application to a high-dimensional data set obtained from an RNA-Seq investigation demonstrates the practical utility of the method.     

Some extensions of zero-inflated models and Bayesian tests for them

Zero-inflated power series distribution is commonly used for modelling count data with extra zeros. Inflation at point zero has been investigated and several tests for zero inflation have been examined. However sometimes, inflation occurs at a point apart from zero. In this case, we say inflation occurs at an arbitrary point j. The j-inflation has been discussed less than zero inflation. In this paper, inflation at an arbitrary point j is studied with more details and a Bayesian test for detecting inflation at point j is presented. The Bayesian method is extended to inflation at arbitrary points i and j. The relationship between the distribution for inflation at point j, inflation at points i and j and missing value imputation is studied. It is shown how to obtain a proper estimate of the population variance if a mean-imputed missing at random data set is used. Some simulation studies are conducted and the proposed Bayesian test is applied on two real data sets.     

Reference Priors for Poisson Processes Involving Nuisance Parameters

A Bayesian reference analysis for determining the posterior distribution of the strength of a radiation source is performed. The only pieces of information available are the numbers of counts gathered in a gross and a background measurement along with the respective counting times and a state-of-knowledge distribution for the efficiency. This situation is addressed by combining the calculations of a “one-at-a-time” reference prior and a reference prior with partial information. The posterior distribution of the source strength obtained with the reference prior leads to credible intervals that have better frequentist coverage than corresponding intervals founded on uniform or Jeffreys’ priors.     

Empirical likelihood estimation for linear regression models with AR(p) error terms with numerical examples

Linear regression models are useful statistical tools to analyze data sets in different fields. There are several methods to estimate the parameters of a linear regression model. These methods usually perform under normally distributed and uncorrelated errors. If error terms are correlated the Conditional Maximum Likelihood (CML) estimation method under normality assumption is often used to estimate the parameters of interest. The CML estimation method is required a distributional assumption on error terms. However, in practice, such distributional assumptions on error terms may not be plausible. In this paper, we propose to estimate the parameters of a linear regression model with autoregressive error term using Empirical Likelihood (EL) method, which is a distribution free estimation method. A small simulation study is provided to evaluate the performance of the proposed estimation method over the CML method. The results of the simulation study show that the proposed estimators based on EL method are remarkably better than the estimators obtained from CML method in terms of mean squared errors (MSE) and bias in almost all the simulation configurations. These findings are also confirmed by the results of the numerical and real data examples.     

Exact and Monte Carlo calculations of integrated likelihoods for the latent class model

The latent class model or multivariate multinomial mixture is a powerful approach for clustering categorical data. It uses a conditional independence assumption given the latent class to which a statistical unit is belonging. In this paper, we exploit the fact that a fully Bayesian analysis with Jeffreys non-informative prior distributions does not involve technical difficulty to propose an exact expression of the integrated complete-data likelihood, which is known as being a meaningful model selection criterion in a clustering perspective. Similarly, a Monte Carlo approximation of the integrated observed-data likelihood can be obtained in two steps: an exact integration over the parameters is followed by an approximation of the sum over all possible partitions through an importance sampling strategy. Then, the exact and the approximate criteria experimentally compete, respectively, with their standard asymptotic BIC approximations for choosing the number of mixture components. Numerical experiments on simulated data and a biological example highlight that asymptotic criteria are usually dramatically more conservative than the non-asymptotic presented criteria, not only for moderate sample sizes as expected but also for quite large sample sizes. This research highlights that asymptotic standard criteria could often fail to select some interesting structures present in the data.     

Bayesian analysis of a linear mixed model with AR(p) errors via MCMC

We develop Bayesian procedures to make inference about parameters of a statistical design with autocorrelated error terms. Modelling treatment effects can be complex in the presence of other factors such as time; for example in longitudinal data. In this paper, Markov chain Monte Carlo methods (MCMC), the Metropolis-Hastings algorithm and Gibbs sampler are used to facilitate the Bayesian analysis of real life data when the error structure can be expressed as an autoregressive model of order p. We illustrate our analysis with real data.     

Fully Bayesian logistic regression with hyper-LASSO priors for high-dimensional feature selection

Feature selection arises in many areas of modern science. For example, in genomic research, we want to find the genes that can be used to separate tissues of different classes (e.g. cancer and normal). One approach is to fit regression/classification models with certain penalization. In the past decade, hyper-LASSO penalization (priors) have received increasing attention in the literature. However, fully Bayesian methods that use Markov chain Monte Carlo (MCMC) for regression/classification with hyper-LASSO priors are still in lack of development. In this paper, we introduce an MCMC method for learning multinomial logistic regression with hyper-LASSO priors. Our MCMC algorithm uses Hamiltonian Monte Carlo in a restricted Gibbs sampling framework. We have used simulation studies and real data to demonstrate the superior performance of hyper-LASSO priors compared to LASSO, and to investigate the issues of choosing heaviness and scale of hyper-LASSO priors.     

Bayesian forecasting for AR(1) models with normal coefficients

This paper presents a Bayesian solution to the problem of time series forecasting, for the case in which the generating process is an autoregressive of order one, with a normal random coefficient. The proposed procedure is based on the predictive density of the future observation. Conjugate priors are used for some parameters, while improper vague priors are used for others.     

Testing Normality for Linear AR( p ) Models

Abstract

This paper proposes a nonparametric mixed test for normality of linear autoregressive time series. The test is based on the best one-step forecast in mean square with time reverse. The test statistic is the mixture of a goodness of fit statistic and Cramer–Von Mises statistic. Some asymptotic properties are developed for the test. Simulated results have shown that the test is easy to use and has good powers. Three examples of applying the test to real data are also included.     

Efficient estimation of auto-regression parameters and innovation distributions for semiparametric integer-valued AR(p) models

Summary.  Integer-valued auto-regressive (INAR) processes have been introduced to model non-negative integer-valued phenomena that evolve over time. The distribution of an INAR( p ) process is essentially described by two parameters: a vector of auto-regression coefficients and a probability distribution on the non-negative integers, called an immigration or innovation distribution. Traditionally, parametric models are considered where the innovation distribution is assumed to belong to a parametric family. The paper instead considers a more realistic semiparametric INAR( p ) model where there are essentially no restrictions on the innovation distribution. We provide an (semiparametrically) efficient estimator of both the auto-regression parameters and the innovation distribution.     

Generating synthetic data to produce public-use microdata for small geographic areas based on complex sample survey data with application to the National Health Interview Survey

Small area statistics obtained from sample survey data provide a critical source of information used to study health, economic, and sociological trends. However, most large-scale sample surveys are not designed for the purpose of producing small area statistics. Moreover, data disseminators are prevented from releasing public-use microdata for small geographic areas for disclosure reasons; thus, limiting the utility of the data they collect. This research evaluates a synthetic data method, intended for data disseminators, for releasing public-use microdata for small geographic areas based on complex sample survey data. The method replaces all observed survey values with synthetic (or imputed) values generated from a hierarchical Bayesian model that explicitly accounts for complex sample design features, including stratification, clustering, and sampling weights. The method is applied to restricted microdata from the National Health Interview Survey and synthetic data are generated for both sampled and non-sampled small areas. The analytic validity of the resulting small area inferences is assessed by direct comparison with the actual data, a simulation study, and a cross-validation study.