Reference Priors Research Articles

Improved Bayesian Inferences for Right-Censored Birnbaum–Saunders Data

This work focuses on making Bayesian inferences for the two-parameter Birnbaum–Saunders (BS) distribution in the presence of right-censored data. A flexible Gibbs sampler is employed to handle the censored BS data in this Bayesian work that relies on Jeffrey’s and Achcar’s reference priors. A comprehensive simulation study is conducted to compare estimates under various parameter settings, sample sizes, and levels of censoring. Further comparisons are drawn with real-world examples involving Type-II, progressively Type-II, and randomly right-censored data. The study concludes that the suggested Gibbs sampler enhances the accuracy of Bayesian inferences, and both the amount of censoring and the sample size are identified as influential factors in such analyses.

A General Scheme for Deriving Conditional Reference Priors

Reference prior algorithms are primarily available for models that have asymptotic posterior normality (i.e., regular case), barring for a specific class of models whose posteriors are not asymptotically normal (i.e., non-regular case). In particular, the current reference prior methodology is contingent on models belonging to regular or non-regular cases. This highlights that the existing reference prior theory lacks a unified approach. A partial breakthrough in unifying the reference prior theory came with Berger et al. (2009) deriving an explicit form of reference prior for single group models (i.e., models with a scalar parameter or models that have all the parameters on equal footing or importance). Unfortunately, their approach does not generalize to multi-group models (i.e., models that have parameters’ subsets ordered according to their importance). In this paper, we show that their main result can be extended with modifications to multi-group models. As a consequence, we obtain a general scheme for deriving conditional reference priors. We also prove that the invariance property of reference priors under particular transformations hold for both regular and non-regular cases. We show the usefulness of our approach by computing reference priors for models that have no known reference priors.

A comparison of objective priors for Cronbach’s coefficient alpha using a balanced random effects model

. In this article, the reference and probability matching priors for Cronbach’s alpha will be derived. The performance of these two priors will be compared to that of the well-known Jeffreys prior and a divergence prior. Cronbach’s alpha is a measure used to assess the reliability of a set of test items. A simulation study will be considered to compare the performance of the priors, where the coverage rates, average interval lengths, and standard deviations of the interval lengths will be computed. A second simulation study will be considered where the mean relative error will be compared for the various priors using three different loss functions. The following loss functions will be considered, Squared error loss, Absolute error loss, and Linex loss. An illustrative example will also be considered. Throughout the article, the random effects approach will be used.

Bayesian Analysis of Unit Log-Logistic Distribution Using Non-Informative Priors

The unit log-logistic distribution is a suitable choice for modeling data enclosed within the unit interval. In this paper, estimating the parameters of the unit-log-logistic distribution is performed through a Bayesian approach with non-informative priors. Specifically, we use Jeffreys, reference, and matching priors, with the latter depending on the interest parameter. We derive the corresponding posterior distributions and validate their propriety. The Bayes estimators are then computed using Markov Chain Monte Carlo techniques. To assess the finite sample performance of these Bayes estimators, we conduct Monte Carlo simulations, evaluating their mean squared errors and their coverage probabilities of the highest posterior density credible intervals. Finally, we use these priors to obtain estimations and credible sets for the parameters in an example of a real data set for illustrative purposes.

Bayesian analysis for the Shannon entropy of the Lomax distribution using noninformative priors

The Lomax distribution is one of the well-known distributions that is used to fit heavy-tailed data. In this paper, we investigate the estimation of Shannon entropy of the Lomax distribution using noninformative priors. Some important priors including Jeffreys prior, reference priors and probability matching priors are presented. We demonstrate that the reference and matching priors for the Lomax entropy do not match the reference and probability matching priors of the Lomax parameters, regardless of which parameter of interest is considered. The propriety and the existence of the expectation of the posterior under each prior are validated, respectively. A simulation study is conducted to assess the frequentist performance of the proposed Bayesian estimates in terms of the mean squared error and coverage probability. Finally, the approach is applied to three real data sets for illustrative purposes.

A multidimensional objective prior distribution from a scoring rule

The construction of objective priors is, at best, challenging for multidimensional parameter spaces. A common practice is to assume independence and set up the joint prior as the product of marginal distributions obtained via “standard” objective methods, such as Jeffreys or reference priors. However, the assumption of independence a priori is not always reasonable, and whether it can be viewed as strictly objective is still open to discussion. In this paper, by extending a previously proposed objective approach based on scoring rules for the one dimensional case, we propose a novel objective prior for multidimensional parameter spaces which yields a dependence structure. The proposed prior has the appealing property of being proper and does not depend on the chosen model; only on the parameter space considered.

Inference for the Johnson SB distribution

This article develops inferential procedures for the Johnson SB distribution and the associated stress-strength model. The maximum likelihood estimators, uniformly minimum variance unbiased estimators and generalized confidence intervals for the parameters of the Johnson SB distribution are presented. The objective Bayesian inference methods based on the Jeffreys and reference priors are derived. Moreover, the generalized confidence lower limits and Bayesian credible lower limits for the reliability of the stress-strength model are provided. The performance of the proposed methods is evaluated by Monte Carlo simulation. Finally, an example is used to illustrate the proposed procedures.

Objective Bayesian analysis of Marshall-Olkin bivariate Weibull distribution with partial information

ABSTRACT In competing risks problem, a subset of risks is needed more attention for inferential purposes. In the objective Bayesian paradigm, reference priors enable to achieve such inferential objectives. In this article, the Marshall-Olkin bivariate Weibull distribution is considered to model the competing risks data. In the availability of partial information for some of the parameters, the reference priors are derived as per the importance of the parameters. The Dirichlet prior is taken as a conditional subjective prior and the marginal reference prior has been derived. Also, the propriety of the resulting posterior density has been proved. The Bayesian estimates of the parameters are obtained under squared error and linear-exponential loss functions. Further, the derived reference prior is used for the computation of Bayes factors or posterior odds in testing the hypothesis that the competing risks are identical. The performance of established Bayesian estimators is illustrated using the Diabetic Retinopathy Study (DRS) and Prostate Cancer data sets. Finally, the model compatibility is done for the considered data sets under Bayesian Paradigm.

Bootstrap confidence intervals of process capability indices Spmk, Spmkc and Cs for Frechet distribution

AbstractIn this paper, we considered the Bayesian estimators under reference and Jeffery's priors and maximum likelihood estimators to estimate the unknown parameters of the process capability indices Spmk, Spmkc, and Cs forFrechetdistribution. Further, we developed bootstrap confidence intervals for aforementioned process capability indices based on above‐mentioned estimators. Monte Carlo simulations are performed to investigate the performance of process capability indices through skewness, kurtosis, mean square error and widths of bootstrap confidence intervals for small, moderate, and large sample sizes. Simulations results indicate that the Bayesian estimator under reference prior outperforms even in small sample sizes, and all performed equally well for larger sample sizes. Moreover, the average width for bootstrap confidence interval for Cs is least in all. Finally, real data is analyzed for illustration purposes.

Hellinger Information Matrix and Hellinger Priors

Hellinger information as a local characteristic of parametric distribution families was first introduced in 2011. It is related to the much older concept of the Hellinger distance between two points in a parametric set. Under certain regularity conditions, the local behavior of the Hellinger distance is closely connected to Fisher information and the geometry of Riemann manifolds. Nonregular distributions (non-differentiable distribution densities, undefined Fisher information or denisities with support depending on the parameter), including uniform, require using analogues or extensions of Fisher information. Hellinger information may serve to construct information inequalities of the Cramer-Rao type, extending the lower bounds of the Bayes risk to the nonregular case. A construction of non-informative priors based on Hellinger information was also suggested by the author in 2011. Hellinger priors extend the Jeffreys rule to nonregular cases. For many examples, they are identical or close to the reference priors or probability matching priors. Most of the paper was dedicated to the one-dimensional case, but the matrix definition of Hellinger information was also introduced for higher dimensions. Conditions of existence and the nonnegative definite property of Hellinger information matrix were not discussed. Hellinger information for the vector parameter was applied by Yin et al. to problems of optimal experimental design. A special class of parametric problems was considered, requiring the directional definition of Hellinger information, but not a full construction of Hellinger information matrix. In the present paper, a general definition, the existence and nonnegative definite property of Hellinger information matrix is considered for nonregular settings.

Abstract WP117: Effectiveness Of Minimally Invasive Surgery With Thrombolysis For Evacuation Of Intracerebral Hemorrhage: Post Hoc Bayesian Analysis Of A Phase 3 Randomized Controlled Trial

Introduction: Bayesian analyses provide meaningful interpretations of clinical trial data in terms of probabilities of treatment effect. This extends the value of results beyond that obtained from traditional binary cut offs. Such interpretations provide actionable information for conditions with high mortality and no proven treatments, such as intracerebral hemorrhage (ICH). Methods: The minimally invasive surgery with thrombolysis in ICH evacuation (MISTIE 3) trial randomized 506 adults with spontaneous, supratentorial ICH of ≥ 30mL to receive either MISTIE intervention or standard medical care. Using reference priors (minimally informative, enthusiastic, skeptical) and a MISTIE 2-derived prior, we estimated probabilities that the effect of MISTIE intervention exceeds pre-specified thresholds of relative risks (RR) for achieving a 365-day Modified Rankin Scale (mRS) score of 0-3. We also computed the effects of MISTIE intervention on 180 and 30-day mRS and 365, 180 and 30-day mortality. Results: Using a minimally informative prior, the posterior probability of MISTIE intervention having any beneficial effect (RR > 1) on 365-day mRS was 70%. The probabilities of observing RR > 1.02 and > 1.10 were 63% and 34%, respectively. The probabilities of observing RR > 1, 1.02 and 1.10, with other priors are: 87%, 82% and 55% (enthusiastic); 68%, 60% and 30% (skeptical); and 73%, 66% and 38% (MISTIE 2-derived). The probabilities that MISTIE intervention has any beneficial effect on 180 and 30-day mRS, and 365, 180, and 30-day mortality are 65%,80%, 93%, 98% and 99%, respectively. RR, 95% credible intervals across various priors are shown in Figure. Conclusions: The probability that the MISTIE intervention has a favorable functional outcome at 365-days among ICH patients is 68 to 87%. Bayesian-derived probabilities of treatment effect may facilitate shared decision making in ICH management, above and beyond p-value-based frequentist interpretation of trial results.

Objective Bayesian Approach for Binary Hypothesis Testing of Multivariate Gaussian Observations

This paper proposes an objective Bayesian detector for the binary hypothesis testing problem in which the observation vectors are IID and follow multivariate Gaussian distribution under both the null and alternative hypotheses. The mean vector and covariance matrix under each hypothesis are not known, they also do not have special structure other than the covariance matrices are partially ordered in the positive definite cone with their difference positive semi-definite. An example of such a problem is the detection of Gaussian random signal in Gaussian noise. Non-informative priors are imposed on the unknown parameters for computing the marginals and evaluating the Bayes factor test statistic, where the prior for the covariance matrix is applied to the Cholesky factorization of the precision matrix. For the null hypothesis, we propose conjugate priors for the unknowns. For the alternative hypothesis, we propose uniform prior on the mean vector, and a class of objective priors that encompasses the Jeffreys, Independence Jeffreys, Geisser and Cornfield, Haar Measure and Reference priors for the precision matrix. The proposed priors enable closed-form expressions for the marginals and lead to an objective Bayes factor having the posteriors mainly governed by the data. The proposed detector is analyzed for justifying the priors used, deriving the theoretical moments, and approximating the false alarm and detection probability densities by the shifted Gamma distributions. The developed detector exhibits improvement in detection performance over the energy detector (ED) and the generalized likelihood ratio test (GLRT).

Objective Prior Distributions to Estimate the Parameters of the Poisson-Exponential Distribution

In this paper, a set of important objective priors are examined for the Bayesian estimation of the parameters present in the Poisson-Exponential distribution PE. We derived the multivariate Jeffreys prior and the Maximal Data Information Prior. Reference prior and others priors proposed in the literature are also analyzed. We show that the posterior densities resulting from these approaches are proper although the respective priors are improper. Monte Carlo simulations are used to compare the efficiencies and to assess the sensitivity of the choice of the priors, mainly for small sample sizes. This simulation study shows that the mean square error, mean bias and coverage probability of credible intervals under Gamma, Jeffreys' rule and Box & Tiao priors presented equal results, whereas Jeffreys and Reference priors showed the best results. The MDIP prior had a worse performance in all analyzed situations showing not to be indicated for Bayesian analysis of the PE distribution. A real data set is analyzed for illustrative purpose of the Bayesian approaches.

Objective Bayesian analysis of the ratio of bivariate lognormal means

In this study, we consider the objective Bayesian analysis for the ratio of means in the bivariate lognormal distribution, in which properly assigning priors for the ratio of means is challenging due to the presence of nuisance parameters. As a result, we create probability matching priors and reference priors for the ratio of means. Jeffreys’ prior and the two group reference prior do not satisfy a first-order matching criterion. We also investigate the posterior distribution property for the general prior class, which includes Jeffreys’ prior, the reference prior, and the matching prior. Through a simulation study, we check the frequentist coverage probabilities and compare them with the generalized confidence interval method. It demonstrated that the proposed probability matching priors match very well with the target coverage probabilities even when the sample sizes are small in a frequentist sense. Two real examples are also provided.

Objective Bayesian Estimation for the Differential Entropy Measure Under Generalized Half-Normal Distribution.

The evaluation of the information entropy content in the data analysis is an effective role in the assessment of fatigue damage. Due to the connection between the generalized half-normal distribution and fatigue extension, the objective inference for the differential entropy of the generalized half-normal distribution is considered in this paper. The Bayesian estimates and associated credible intervals are discussed based on different non-informative priors including Jeffery, reference, probability matching, and maximal data information priors for the differential entropy measure. The Metropolis-Hastings samplers data sets are used to estimate the posterior densities and then compute the Bayesian estimates. For comparison purposes, the maximum likelihood estimators and asymptotic confidence intervals of the differential entropy are derived. An intensive simulation study is conducted to evaluate the performance of the proposed statistical inference methods. Two real data sets are analyzed by the proposed methodology for illustrative purposes as well. Finally, non-informative priors for the original parameters of generalized half-normal distribution based on the direct and transformation of the entropy measure are also proposed and compared.

Bayesian analysis for the Lomax model using noninformative priors

The Lomax distribution is an important member in the distribution family. In this paper, we systematically develop an objective Bayesian analysis of data from a Lomax distribution. Noninformative priors, including probability matching priors, the maximal data information (MDI) prior, Jeffreys prior and reference priors, are derived. The propriety of the posterior under each prior is subsequently validated. It is revealed that the MDI prior and one of the reference priors yield improper posteriors, and the other reference prior is a second-order probability matching prior. A simulation study is conducted to assess the frequentist performance of the proposed Bayesian approach. Finally, this approach along with the bootstrap method is applied to a real data set.

Application of objective priors for the multivariate Lomax distribution

For a model incorporating the effect of a common environment on several components of a system, a multivariate Lomax distribution (MLD) is generally considered by mixing exponential variables. Objective Bayesian has very good frequentist properties and provides a moderate solution for the prior elicitation which is one of important and difficult issues on Bayesian analysis. In this paper, we develop noninformative priors, such as the probability matching priors and reference priors, for the parameters of the MLD. We proved that a reference prior for the shape parameter is a first-order probability matching prior, but the reference priors for the scale parameters do not satisfy the first-order matching criterion. In addition, a second-order probability matching prior does not exist for all parameters. We also presented the conditions that make the posterior distributions for the general prior, including the probability matching prior and reference priors, to be proper. In particular, Jeffreys’ prior and probability matching priors for all parameters give proper posteriors, whereas reference priors for scale parameters give improper posteriors.

Statistical Analysis for Competing Risks' Model with Two Dependent Failure Modes from Marshall-Olkin Bivariate Gompertz Distribution.

The bivariate or multivariate distribution can be used to account for the dependence structure between different failure modes. This paper considers two dependent competing failure modes from Gompertz distribution, and the dependence structure of these two failure modes is handled by the Marshall–Olkin bivariate distribution. We obtain the maximum likelihood estimates (MLEs) based on classical likelihood theory and the associated bootstrap confidence intervals (CIs). The posterior density function based on the conjugate prior and noninformative (Jeffreys and Reference) priors are studied; we obtain the Bayesian estimates in explicit forms and construct the associated highest posterior density (HPD) CIs. The performance of the proposed methods is assessed by numerical illustration.

The Role of Prior Probabilities on Parameter Estimation in Hydrological Models

AbstractBayes theorem provides a formal framework for combining prior and sample information for parameter estimation in the presence of measurement and structural errors. Prior knowledge, however, may not always be available or may be too vague to incorporate into a prior distribution. In such cases, a reference prior must be chosen for an objective analysis. Typically, a uniform density over the possible ranges of parameters is chosen as the reference prior. However, the validity of a uniform prior as a reference prior is seldom questioned. In this study, an information‐theoretic approach is pursued to derive reference priors, and the results are compared to those obtained by using a uniform prior. Examples of estimating saturated hydraulic conductivity are presented. Priors over hydraulic conductivity obtained by using the information‐theoretic approach are transformation‐invariant and typically nonuniform. The choice between information‐theoretic and uniform prior influences the posterior distribution of hydraulic conductivity, when sample information is small. The use of reference prior is also demonstrated through the PDG‐GIUH hydrologic model, and issues of computational tractability are addressed.

Approximate reference priors for Gaussian random fields

AbstractReference priors are theoretically attractive for the analysis of geostatistical data since they enable automatic Bayesian analysis and have desirable Bayesian and frequentist properties. But their use is hindered by computational hurdles that make their application in practice challenging. In this work, we derive a new class of default priors that approximate reference priors for the parameters of some Gaussian random fields. It is based on an approximation to the integrated likelihood of the covariance parameters derived from the spectral approximation of stationary random fields. This prior depends on the structure of the mean function and the spectral density of the model evaluated at a set of spectral points associated with an auxiliary regular grid. In addition to preserving the desirable Bayesian and frequentist properties, these approximate reference priors are more stable, and their computations are much less onerous than those of exact reference priors. Unlike exact reference priors, the marginal approximate reference prior of correlation parameter is always proper, regardless of the mean function or the smoothness of the correlation function. This property has important consequences for covariance model selection. An illustration comparing default Bayesian analyses is provided with a dataset of lead pollution in Galicia, Spain.