Confidence intervals,significance values,maximum likelihood estimates,etc. sharpened into Occam’s razors

Abstract

Confidence sets, p values, maximum likelihood estimates, and other results of non-Bayesian statistical methods may be adjusted to favor sampling distributions that are simple compared to others in the parametric family. The adjustments are derived from a prior likelihood function previously used to adjust posterior distributions.     

BAYESIAN SUBSET SELECTION AND MODEL AVERAGING USING A CENTRED AND DISPERSED PRIOR FOR THE ERROR VARIANCE

This article proposes a new data‐based prior distribution for the error variance in a Gaussian linear regression model, when the model is used for Bayesian variable selection and model averaging. For a given subset of variables in the model, this prior has a mode that is an unbiased estimator of the error variance but is suitably dispersed to make it uninformative relative to the marginal likelihood. The advantage of this empirical Bayes prior for the error variance is that it is centred and dispersed sensibly and avoids the arbitrary specification of hyperparameters. The performance of the new prior is compared to that of a prior proposed previously in the literature using several simulated examples and two loss functions. For each example our paper also reports results for the model that orthogonalizes the predictor variables before performing subset selection. A real example is also investigated. The empirical results suggest that for both the simulated and real data, the performance of the estimators based on the prior proposed in our article compares favourably with that of a prior used previously in the literature.     

Bayesian analysis of elapsed times in continuous‐time Markov chains

The authors consider Bayesian analysis for continuous‐time Markov chain models based on a conditional reference prior. For such models, inference of the elapsed time between chain observations depends heavily on the rate of decay of the prior as the elapsed time increases. Moreover, improper priors on the elapsed time may lead to improper posterior distributions. In addition, an infinitesimal rate matrix also characterizes this class of models. Experts often have good prior knowledge about the parameters of this matrix. The authors show that the use of a proper prior for the rate matrix parameters together with the conditional reference prior for the elapsed time yields a proper posterior distribution. The authors also demonstrate that, when compared to analyses based on priors previously proposed in the literature, a Bayesian analysis on the elapsed time based on the conditional reference prior possesses better frequentist properties. The type of prior thus represents a better default prior choice for estimation software.     

The empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior under Stein's loss function

For the hierarchical Poisson and gamma model, we calculate the Bayes posterior estimator of the parameter of the Poisson distribution under Stein's loss function which penalizes gross overestimation and gross underestimation equally and the corresponding Posterior Expected Stein's Loss (PESL). We also obtain the Bayes posterior estimator of the parameter under the squared error loss and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior by two methods. In numerical simulations, we have illustrated: The two inequalities of the Bayes posterior estimators and the PESLs; the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters; the goodness-of-fit of the model to the simulated data. The numerical results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, we exploit the attendance data on 314 high school juniors from two urban high schools to illustrate our theoretical studies.     

Nonparametric Bayesian inference for multivariate density functions using Feller priors

Multivariate density estimation plays an important role in investigating the mechanism of high-dimensional data. This article describes a nonparametric Bayesian approach to the estimation of multivariate densities. A general procedure is proposed for constructing Feller priors for multivariate densities and their theoretical properties as nonparametric priors are established. A blocked Gibbs sampling algorithm is devised to sample from the posterior of the multivariate density. A simulation study is conducted to evaluate the performance of the procedure.     

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.     

Bayesian bridge-randomized penalized quantile regression estimation for linear regression model with AP(q) perturbation

Bridge penalized regression has many desirable statistical properties such as unbiasedness, sparseness as well as ‘oracle’. In Bayesian framework, bridge regularized penalty can be implemented based on generalized Gaussian distribution (GGD) prior. In this paper, we incorporate Bayesian bridge-randomized penalty and its adaptive version into the quantile regression (QR) models with autoregressive perturbations to conduct Bayesian penalization estimation. Employing the working likelihood of the asymmetric Laplace distribution (ALD) perturbations, the Bayesian joint hierarchical models are established. Based on the mixture representations of the ALD and generalized Gaussian distribution (GGD) priors of coefficients, the hybrid algorithms based on Gibbs sampler and Metropolis-Hasting sampler are provided to conduct fully Bayesian posterior estimation. Finally, the proposed Bayesian procedures are illustrated by some simulation examples and applied to a real data application of the electricity consumption.     

Bayesian approach to epsilon-skew-normal family

The estimation problem of epsilon-skew-normal (ESN) distribution parameters is considered within Bayesian approaches. This family of distributions contains the normal distribution, can be used for analyzing the asymmetric and near-normal data. Bayesian estimates under informative and non informative Jeffreys prior distributions are obtained and performances of ESN family and these estimates are shown via a simulation study. A real data set is also used to illustrate the ideas.     

Bayesian analysis of a multiple-recapture model

In the Bayesian analysis of a multiple-recapture census, different diffuse prior distributions can lead to markedly different inferences about the population size N. Through consideration of the Fisher information matrix it is shown that the number of captures in each sample typically provides little information about N. This suggests that if there is no prior information about capture probabilities, then knowledge of just the sample sizes and not the number of recaptures should leave the distribution of Nunchanged. A prior model that has this property is identified and the posterior distribution is examined. In particular, asymptotic estimates of the posterior mean and variance are derived. Differences between Bayesian and classical point and interval estimators are illustrated through examples.     

Addressing potential prior‐data conflict when using informative priors in proof‐of‐concept studies

Bayesian methods are increasingly used in proof‐of‐concept studies. An important benefit of these methods is the potential to use informative priors, thereby reducing sample size. This is particularly relevant for treatment arms where there is a substantial amount of historical information such as placebo and active comparators. One issue with using an informative prior is the possibility of a mismatch between the informative prior and the observed data, referred to as prior‐data conflict. We focus on two methods for dealing with this: a testing approach and a mixture prior approach. The testing approach assesses prior‐data conflict by comparing the observed data to the prior predictive distribution and resorting to a non‐informative prior if prior‐data conflict is declared. The mixture prior approach uses a prior with a precise and diffuse component. We assess these approaches for the normal case via simulation and show they have some attractive features as compared with the standard one‐component informative prior. For example, when the discrepancy between the prior and the data is sufficiently marked, and intuitively, one feels less certain about the results, both the testing and mixture approaches typically yield wider posterior‐credible intervals than when there is no discrepancy. In contrast, when there is no discrepancy, the results of these approaches are typically similar to the standard approach. Whilst for any specific study, the operating characteristics of any selected approach should be assessed and agreed at the design stage; we believe these two approaches are each worthy of consideration. Copyright © 2015 John Wiley & Sons, Ltd.