Frequentist Coverage Research Articles

Commentary: Priors, parameters, and probability: a bayesian perspective on sensitivity analysis.

Commentary: Priors, parameters, and probability: a bayesian perspective on sensitivity analysis.

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.

Objective Bayes analysis of zero-inflated Poisson distribution with application to healthcare data

In this article, non-informative priors are investigated for a zero-inflated Poisson distribution with two parameters: the probability of zeros and the mean of the Poisson part. Both the reference prior and the Jeffreys prior are derived and shown to be second-order matching priors when only the mean of the Poisson part is of interest. However, when the probability of zeros is of interest, the reference prior is still a second-order matching prior, whereas the Jeffreys prior is not so. Furthermore, when both parameters are of interest, the reference prior is a unique second-order matching prior. Frequentist coverage probabilities of the posterior confidence sets based on the Jeffreys and reference priors are compared with each other using Monte Carlo simulations and with confidence sets based on the maximum likelihood estimation.

Modeling and one-sample Bayes inference for a new distribution for 3-D rotations motivated by a central limit

For modeling orientation data represented as 3 × 3 rotation matrices, we develop a wrapped trivariate normal distribution (wTND) under which random rotations have simple geometric construction as symmetric errors about a mean. While of interest in its own right, the wTND also provides simple and effective approximations to the isotropic Gaussian distribution on rotations, with some advantages over approximations based on other commonly used models. We develop non-informative Bayes inference for the wTND via Markov Chain Monte Carlo methods that allow straightforward computations in a model where maximum likelihood is undefined. Credible regions for model parameters (including a fixed 3× 3 mean rotation) are shown to possess good frequentist coverage properties. We illustrate the model and inference method with orientation data collected in texture analysis from materials science.

Objective Bayesian analysis for autoregressive models with nugget effects

The conditional autoregressive (CAR) and simultaneous autoregressive (SAR) models both have been used extensively for the analysis of spatial structure underlying lattice data in many areas, such as epidemiology, demographics, economics, and geography. Default Bayesian analyses have been conducted recently, but the Bayesian approach has not used or explored these two models with nugget effects. In this paper, we consider general autoregressive models including both CAR and SAR models. The Jeffreys-rule, independence Jeffreys, commonly used reference and “exact” reference priors are derived. The propriety of the marginal priors and joint posteriors is studied for a large class of objective priors. Various Jeffreys and reference priors are shown to yield improper posteriors and only the Jeffreys-rule and the “exact” reference priors yield proper posteriors. We make comparisons for these two objective priors using the frequentist coverage probabilities of the credible intervals. An illustration is given using a real spatial data-set.

Improved confidence intervals for the exponential mean via tail functions

The method of tail functions is applied to confidence estimation of the exponential mean in the presence of prior information. It is shown how the “ordinary” confidence interval can be generalized using a class of tail functions and then engineered for optimality, in the sense of minimizing prior expected length over that class, whilst preserving frequentist coverage. It is also shown how to derive the globally optimal interval, and how to improve on this using tail functions when criteria other than length are taken into consideration. Probabilities of false coverage are reported for some of the intervals under study, and the theory is illustrated by application to confidence estimation of a reliability coefficient based on some survival data.

Nonparametric Bernstein–von Mises theorems in Gaussian white noise

Bernstein–von Mises theorems for nonparametric Bayes priors in the Gaussian white noise model are proved. It is demonstrated how such results justify Bayes methods as efficient frequentist inference procedures in a variety of concrete nonparametric problems. Particularly Bayesian credible sets are constructed that have asymptotically exact $1-\alpha$ frequentist coverage level and whose $L^{2}$-diameter shrinks at the minimax rate of convergence (within logarithmic factors) over Hölder balls. Other applications include general classes of linear and nonlinear functionals and credible bands for auto-convolutions. The assumptions cover nonconjugate product priors defined on general orthonormal bases of $L^{2}$ satisfying weak conditions.

A note on approximate Bayesian credible sets based on modified loglikelihood ratios

Higher-order asymptotic arguments for a scalar parameter of interest have been widely investigated for Bayesian inference. In this paper the theory of asymptotic expansions is discussed for a vector parameter of interest. A modified loglikelihood ratio is suggested, which can be used to derive approximate Bayesian credible sets with accurate frequentist coverage. Three examples are illustrated.

Objective Bayesian analysis of spatial models with separable correlation functions

AbstractThis paper considers general linear models for Gaussian geostatistical data with multi‐dimensional separable correlation functions involving multiple parameters. We derive various objective priors, such as the Jeffreys‐rule, independence Jeffreys, and usual and exact reference priors for the model parameters. In addition, we relax and simplify the assumptions in Paulo (2005) for the propriety of the posteriors in the general setup. We show that the frequentist coverage of posterior credible intervals for a function of range parameters do not depend on the regression coefficient or error variance. These objective priors and a proper flat prior based on ML estimates are compared by examining the frequentist coverage of equal‐tailed Bayesian credible intervals. An illustrative example is given from the field of complex computer model validations. The Canadian Journal of Statistics 41: 488–507; 2013 © 2013 Statistical Society of Canada

Empirical Bayes Confidence Intervals for Selected Parameters in High-Dimensional Data

Modern statistical problems often involve a large number of populations and hence a large number of parameters that characterize these populations. It is common for scientists to use data to select the most significant populations, such as those with the largest t statistics. The scientific interest often lies in studying and making inferences regarding these parameters, called the selected parameters, corresponding to the selected populations. The current statistical practices either apply a traditional procedure assuming there were no selection—a practice that is not valid—or they use the Bonferroni-type procedure that is valid but very conservative and often noninformative. In this article, we propose valid and sharp confidence intervals that allow scientists to select parameters and to make inferences for the selected parameters based on the same data. This type of confidence interval allows the users to zero in on the most interesting selected parameters without collecting more data. The validity of confidence intervals is defined as the controlling of Bayes coverage probability so that it is no less than a nominal level uniformly over a class of prior distributions for the parameter. When a mixed model is assumed and the random effects are the key parameters, this validity criterion is exactly the frequentist criterion, since the Bayes coverage probability is identical to the frequentist coverage probability. Assuming that the observations are normally distributed with unequal and unknown variances, we select parameters with the largest t statistics. We then construct sharp empirical Bayes confidence intervals for these selected parameters, which have either a large Bayes coverage probability or a small Bayes false coverage rate uniformly for a class of priors. Our intervals, applicable to any high-dimensional data, are applied to microarray data and are shown to be better than all the alternatives. It is also anticipated that the same intervals would be valid for any selection rule. Supplementary materials for this article are available online.

A Bayesian Approach for Estimating Mediation Effects With Missing Data

Methodologists have developed mediation analysis techniques for a broad range of substantive applications, yet methods for estimating mediating mechanisms with missing data have been understudied. This study outlined a general Bayesian missing data handling approach that can accommodate mediation analyses with any number of manifest variables. Computer simulation studies showed that the Bayesian approach produced frequentist coverage rates and power estimates that were comparable to those of maximum likelihood with the bias-corrected bootstrap. We share an SAS macro that implements Bayesian estimation and use 2 data analysis examples to demonstrate its use.

Bayesian Recovery of the Initial Condition for the Heat Equation

We study a Bayesian approach to recovering the initial condition for the heat equation from noisy observations of the solution at a later time. We consider a class of prior distributions indexed by a parameter quantifying “smoothness” and show that the corresponding posterior distributions contract around the true parameter at a rate that depends on the smoothness of the true initial condition and the smoothness and scale of the prior. Correct combinations of these characteristics lead to the optimal minimax rate. One type of priors leads to a rate-adaptive Bayesian procedure. The frequentist coverage of credible sets is shown to depend on the combination of the prior and true parameter as well, with smoother priors leading to zero coverage and rougher priors to (extremely) conservative results. In the latter case, credible sets are much larger than frequentist confidence sets, in that the ratio of diameters diverges to infinity. The results are numerically illustrated by a simulated data example.

Objective Bayesian analysis for bivariate Marshall–Olkin exponential distribution

The Bayesian estimators for the unknown parameters of the bivariate Marshall–Olkin exponential distribution under noninformative priors have been considered and several reference priors have been derived. A class of priors is found by matching the coverage probability of one-side Bayesian credible intervals with the corresponding frequentist coverage probabilities. It is noted that some of the reference priors are also matching priors and the posterior distributions based on the reference priors and matching priors are proper. Closed forms of Bayesian estimators are obtained with respect to the quadratic loss function. Gibbs sampling is utilized to obtain the credible intervals and coverage probabilities of parameters. Comparisons in the efficiency of the maximum likelihood estimators and Bayesian estimators under different reference priors and matching priors for various sample sizes have been done by Monte Carlo simulations. A real data set is analyzed for illustrative purpose.

On Bayesian credible sets, restricted parameter spaces and frequentist coverage

For estimating a lower bounded parametric function in the framework of Marchand and Strawderman [6], we provide through a unified approach a class of Bayesian confidence intervals with credibility $1-\alpha$ and frequentist coverage probability bounded below by $\frac{1-\alpha}{1+\alpha}$. In cases where the underlying pivotal distribution is symmetric, the findings represent extensions with respect to the specification of the credible set achieved through the choice of a spending function, and include Marchand and Strawderman’s HPD procedure result. For non-symmetric cases, the determination of a such a class of Bayesian credible sets fills a gap in the literature and includes an “equal-tails” modification of the HPD procedure. Several examples are presented demonstrating wide applicability.

A matching prior based on the modified profile likelihood in a generalized Weibull stress‐strength model

AbstractBayesian inference of a generalized Weibull stress‐strength model (SSM) with more than one strength component is considered. For this problem, properly assigning priors for the reliabilities is challenging due to the presence of nuisance parameters. Matching priors, which are priors matching the posterior probabilities of certain regions with their frequentist coverage probabilities, are commonly used but difficult to derive in this problem. Instead, we apply an alternative method and derive a matching prior based on a modification of the profile likelihood. Simulation studies show that this proposed prior performs well in terms of frequentist coverage and estimation even when the sample sizes are minimal. The prior is applied to two real datasets. The Canadian Journal of Statistics 41: 83–97; 2013 © 2012 Statistical Society of Canada

Objective Bayesian analysis for CAR models

Objective priors, especially reference priors, have been studied extensively for spatial data in the last decade. In this paper, we study objective priors for a CAR model. In particular, the properties of the reference prior and the corresponding posterior are studied. Furthermore, we show that the frequentist coverage probabilities of posterior credible intervals depend only on the spatial dependence parameter \(\rho \), and not on the regression coefficient or the error variance. Based on the simulation study for comparing the reference and Jeffreys priors, the performance of two reference priors is similar and better than the Jeffreys priors. One spatial dataset is used for illustration.

Bayes Inference for a Tractable New Class of Non-symmetric Distributions for 3-Dimensional Rotations

Both existing models for non-symmetric distributions on 3-dimensional rotations and their associated one-sample inference methods have serious limitations in terms of both interpretability and ease of use. Based on the intuitively appealing Uniform Axis- Random Spin (UARS) construction of Bingham, Nordman, and Vardeman (2009) for symmetric families of distributions, we propose new highly interpretable and tractable classes of non-symmetric distributions that are derived from mixing UARS distributions. These have an appealing Preferred Axis-Random Spin (PARS) construction and (unlike existing models) directly interpretable parameters. Non-informative one-sample Bayes inference in these models is a direct generalization of UARS methods introduced in Bingham, Vardeman, and Nordman (2009), where credible levels were found to be essentially equivalent to frequentist coverage probabilities. We apply the new models and inference methods to a problem in biomechanics, where comparison of model parameters provides meaningful comparisons for the nature of movement about the calcaneocuboid joint of three different primate subjects.

Objective Bayesian Analysis of Skew‐tDistributions

Abstract. We study the Jeffreys prior and its properties for the shape parameter of univariate skew‐tdistributions with linear and nonlinear Student'stskewing functions. In both cases, we show that the resulting priors for the shape parameter are symmetric around zero and proper. Moreover, we propose a Student'stapproximation of the Jeffreys prior that makes an objective Bayesian analysis easy to perform. We carry out a Monte Carlo simulation study that demonstrates an overall better behaviour of the maximum aposterioriestimator compared with the maximum likelihood estimator. We also compare the frequentist coverage of the credible intervals based on the Jeffreys prior and its approximation and show that they are similar. We further discuss location‐scale models under scale mixtures of skew‐normal distributions and show some conditions for the existence of the posterior distribution and its moments. Finally, we present three numerical examples to illustrate the implications of our results on inference for skew‐tdistributions.

Coverage Properties of Confidence Intervals for Generalized Additive Model Components

Abstract. We study the coverage properties of Bayesian confidence intervals for the smooth component functions of generalized additive models (GAMs) represented using any penalized regression spline approach. The intervals are the usual generalization of the intervals first proposed by Wahba and Silverman in 1983 and 1985, respectively, to the GAM component context. We present simulation evidence showing these intervals have close to nominal ‘across‐the‐function’ frequentist coverage probabilities, except when the truth is close to a straight line/plane function. We extend the argument introduced byNychka in 1988for univariate smoothing splines to explain these results. The theoretical argument suggests that close to nominal coverage probabilities can be achieved, provided that heavy oversmoothing is avoided, so that the bias is not too large a proportion of the sampling variability. The theoretical results allow us to derive alternative intervals from a purely frequentist point of view, and to explain the impact that the neglect of smoothing parameter variability has on confidence interval performance. They also suggest switching the target of inference for component‐wise intervals away from smooth components in the space of the GAM identifiability constraints.

Non-informative priors in the generalized gamma stress–strength systems

This article deals with non-informative priors for parameters when both stress and strength follow generalized gamma distributions. First, the orthogonal reparameterization is treated and then, using this reparameterization, Jeffreys’ prior, group ordering reference priors, and matching priors are derived. The propriety of posterior distributions is investigated and marginal posterior distributions are provided under those non-informative priors. The question of whether or not the reference priors satisfy the probability matching criterion is addressed. Finally, the reference prior that satisfies the probability matching criterion is shown to be good in the sense of frequentist coverage probability of the posterior quantile. Accepted in 2005 for a special issue on Reliability co-edited by Hoang Pham, Rutgers University; Dong Ho Park, Hallym University, Korea; and Richard Cassady, University of Arkansas.