Reference Priors Research Articles

Objective Bayesian inference for the Capability index of the Weibull distribution and its generalization

The Weibull distribution plays an important role in reliability and quality control monitoring. This model has been widely used to describe the process capability index (PCI) when data do not follow a normal distribution. In this scenario, the current studies focus on estimating the parameters using classical inference. In this paper, we consider Bayesian methods to estimate the PCI denominated Cpk from an objective perspective using reference priors. The proposed inference is further extended to a generalized version of the Weibull distribution that provides a good fit for more complex data with non-monotone hazard behavior. The posterior distributions are constructed and Bayes estimators based on the median are proposed. In this case, Markov Chain Monte Carlo methods are used to achieve the estimates and from an extensive simulation study, we observe that good results are observed in terms of mean relative and squared errors. The proposed approach is also used to construct adequate credibility intervals with low computational cost and accurate coverage probabilities. A real data application is presented which confirms that our proposed approach outperforms the current methods.

The Bayes Estimators of the Variance and Scale Parameters of the Normal Model With a Known Mean for the Conjugate and Noninformative Priors Under Stein's Loss.

For the normal model with a known mean, the Bayes estimation of the variance parameter under the conjugate prior is studied in Lehmann and Casella (1998) and Mao and Tang (2012). However, they only calculate the Bayes estimator with respect to a conjugate prior under the squared error loss function. Zhang (2017) calculates the Bayes estimator of the variance parameter of the normal model with a known mean with respect to the conjugate prior under Stein’s loss function which penalizes gross overestimation and gross underestimation equally, and the corresponding Posterior Expected Stein’s Loss (PESL). Motivated by their works, we have calculated the Bayes estimators of the variance parameter with respect to the noninformative (Jeffreys’s, reference, and matching) priors under Stein’s loss function, and the corresponding PESLs. Moreover, we have calculated the Bayes estimators of the scale parameter with respect to the conjugate and noninformative priors under Stein’s loss function, and the corresponding PESLs. The quantities (prior, posterior, three posterior expectations, two Bayes estimators, and two PESLs) and expressions of the variance and scale parameters of the model for the conjugate and noninformative priors are summarized in two tables. After that, the numerical simulations are carried out to exemplify the theoretical findings. Finally, we calculate the Bayes estimators and the PESLs of the variance and scale parameters of the S&P 500 monthly simple returns for the conjugate and noninformative priors.

A simulation study to compare reference and other priors in the case of a standard univariate Student t-distribution

In this paper, reference and probability-matching priors are derived for the univariate Student t-distribution. These priors generally lead to procedures with properties frequentists can relate to while still retaining Bayes validity. The priors are tested by performing simulation studies. The focus is on the relative mean squared error from the posterior median and on the frequentist coverage of the 95% credibility intervals for a sample of size n = 30. Average interval lengths of the credibility intervals as well as the modes of the interval lengths based on 2 000 simulations are also considered. The performance of the priors is also tested on real data, namely daily logarithmic returns of IBM stocks. Keywords: Coverage percentages, Credibility intervals, Log-returns, Mean squared error, Probability-matching prior, Reference priors

Bayesian control charts for tolerance limits in the case of normal populations

By using air-lead data analysed by Krishnamoorthy and Mathew (2009) a Bayesian procedure is applied to obtain control limits for the upper one-sided tolerance limit. Reference and probability matching priors are derived for the pth quantile of a normal distribution. By simulating the predictive density of a future upper one-sided tolerance limit, "run-lengths" and average "run-lengths" are derived. In the second part of this paper control limits are derived for one-sided tolerance limits for the distribution of the difference between two normal random variables. This article illustrates the flexibility and unique features of the Bayesian simulation method for obtaining the posterior predictive distribution of a future one-sided tolerance limit. Keywords: Control charts, Probability-matching prior, Reference prior, Tolerance limits

Objective bayesian analysis for multiple repairable systems.

This article focus on the analysis of the reliability of multiple identical systems that can have multiple failures over time. A repairable system is defined as a system that can be restored to operating state in the event of a failure. This work under minimal repair, it is assumed that the failure has a power law intensity and the Bayesian approach is used to estimate the unknown parameters. The Bayesian estimators are obtained using two objective priors know as Jeffreys and reference priors. We proved that obtained reference prior is also a matching prior for both parameters, i.e., the credibility intervals have accurate frequentist coverage, while the Jeffreys prior returns unbiased estimates for the parameters. To illustrate the applicability of our Bayesian estimators, a new data set related to the failures of Brazilian sugar cane harvesters is considered.

Objective Bayesian analysis of accelerated degradation models based on Wiener process with correlation

In this paper we propose an objective Bayesian method to analyze the accelerated degradation model based on the new Wiener process stated in Ye, Chen, and Shen (Reliab. Eng. Syst. Safe., 2015, 139: 58–67). The Jeffreys prior and reference priors are derived after reparameterization, and the posterior properties under each prior are validated. A Metropolis random walk algorithm is then employed to generate samples from the posterior distributions, and a simulation study is carried out to reveal the frequentist properties of these priors. Finally, we apply the approach to a real data and obtain the mean-time-to-failure (MTTF) under the usage condition.

Objective priors for common correlation coefficient in bivariate normal populations

Various objective priors have been defined for the common correlation coefficient concerning several bivariate normal populations. In this paper, the proposed approach relies on the asymptotic matching of coverage probabilities corresponding to Bayesian credible intervals considering the corresponding frequentist ones. In the present paper, we focus on several matching criteria including quantile matching, distribution function matching, highest posterior density matching, and matching via inversion of test statistics. In addition, we consider reference priors for different groups of ordering. The proposed methods are investigated and compared between each other in terms of a frequentist coverage probability and then, they are illustrated through a simulation study and two real data examples.

Fast and scalable computations for Gaussian hierarchical models with intrinsic conditional autoregressive spatial random effects

Fast algorithms are developed for Bayesian analysis of Gaussian hierarchical models with intrinsic conditional autoregressive (ICAR) spatial random effects. To achieve computational speed-ups, first a result is proved on the equivalence between the use of an improper CAR prior with centering on the fly and the use of a sum-zero constrained ICAR prior. This equivalence result then provides the key insight for the algorithms, which are based on rewriting the hierarchical model in the spectral domain. The two novel algorithms are the Spectral Gibbs Sampler (SGS) and the Spectral Posterior Maximizer (SPM). Both algorithms are based on one single matrix spectral decomposition computation. After this computation, the SGS and SPM algorithms scale linearly with the sample size. The SGS algorithm is preferable for smaller sample sizes, whereas the SPM algorithm is preferable for sample sizes large enough for asymptotic calculations to provide good approximations. Because the matrix spectral decomposition needs to be computed only once, the SPM algorithm has computational advantages over algorithms based on sparse matrix factorizations (which need to be computed for each value of the random effects variance parameter) in situations when many models need to be fitted. Three simulation studies are performed: the first simulation study shows improved performance in computational speed in estimation of the SGS algorithm compared to an algorithm that uses the spectral decomposition of the precision matrix; the second simulation study shows that for model selection computations with 10 regressors and sample sizes varying from 49 to 3600, when compared to the current fastest state-of-the-art algorithm implemented in the R package INLA, SPM computations are 550 to 1825 times faster; the third simulation study shows that, when compared to default INLA settings, SGS and SPM combined with reference priors provide much more adequate uncertainty quantification. Finally, the application of the novel SGS and SPM algorithms is illustrated with a spatial regression study of county-level median household income for 3108 counties in the contiguous United States in 2017.

Noninformative priors for the ratio of variabilities in a bivariate normal population

In this paper, we consider some objective priors for the ratio of variabilities in a bivariate normal distribution. We develop the first and second order matching priors and reference priors. We obtain that the second order matching prior matches the alternative coverage probabilities up to the same order. It is also an HPD matching priors. It turns out that the derived reference priors do not satisfy a second order matching criterion. The simulation result shows that the second order matching prior performs better than reference priors based on matching the target coverage probabilities in a frequentist sense. Finally, we show that the second order matching prior and reference priors produce confidence sets with an expected length shorter than the Cox and Reid adjustment.

Bayesian beta nonlinear models with constrained parameters to describe ruminal degradation kinetics

The models used to describe the kinetics of ruminal degradation are usually nonlinear models where the dependent variable is the proportion of degraded food. The method of least squares is the standard approach used to estimate the unknown parameters but this method can lead to unacceptable predictions. To solve this issue, a beta nonlinear model and the Bayesian perspective is proposed in this article. The application of standard methodologies to obtain prior distributions, such as the Jeffreys prior or the reference priors, involves serious difficulties here because this model is a nonlinear non-normal regression model, and the constrained parameters appear in the log-likelihood function through the Gamma function. This paper proposes an objective method to obtain the prior distribution, which can be applied to other models with similar complexity, can be easily implemented in OpenBUGS, and solves the problem of unacceptable predictions. The model is generalized to a larger class of models. The methodology was applied to real data with three models that were compared using the Deviance Information Criterion and the root mean square prediction error. A simulation study was performed to evaluate the coverage of the credible intervals.

Objective Bayesian analysis for generalized exponential stress–strength model

In reliability studies, a stress–strength model is often used to analyze a system that fails whenever the applied stress is greater than the strength. Statistical inference of reliability is widely used in a number of areas, such as engineering, clinical trials, and quality control. In addition to the common stress–strength model with one stress and one strength, the reliability of more complex systems has also been studied. In this study, we consider the reliability of a generalized stress–strength model that consists of a serial system with one stress and multiple strengths. We then develop the probability matching priors and reference priors for a generalized exponential stress–strength model. We demonstrate that the two-group reference prior and Jeffreys prior are not a matching prior. Through a simulation study and real data example, we also demonstrate that the proposed probability matching priors match the target coverage probabilities in a frequentist sense even for a small sample size.

Current Frame Priors Assisted Neural Network for Intra Prediction

Intra prediction is the key technology to reduce spatial redundancy in the modern video coding standard. Recently, deep learning based methods that directly generate the intra prediction by neural network achieve superior performance than traditional directional based intra prediction. However, these methods lack the ability to handle complex blocks which contain mixed directional textures or recurrent patterns since they only use the neighboring reference samples of the current one. The other intermediate information denoted as reference priors in this paper generated during the coding process is not exploited. In this paper, a Current Frame Priors assisted Neural Network (CFPNN) is presented to improve the intra prediction efficiency. Specifically, we utilize the local contextual information provided by the neighboring multiple references as the primary inference source. In addition to the neighboring references, we additionally use the other two reference priors within the current frame – the predictor searched by intra block copy (IntraBC) and the corresponding residual component. The IntraBC predictor provides useful nonlocal information to help generate more accurate prediction for complex blocks together with neighboring local information. While the residual component contains unique information that reflects the characteristics of the block to some extent is utilized to reduce the noise contained in the reconstructed reference samples. Moreover, we investigate the best way to integrate the proposed method into the codec. Experimental results demonstrate that compared to HEVC, our proposed CFPNN achieves an average of 4.1% BD-rate reduction for the luma component under the All Intra configuration.

On Default Priors for Robust Bayesian Estimation with Divergences.

This paper presents objective priors for robust Bayesian estimation against outliers based on divergences. The minimum -divergence estimator is well-known to work well in estimation against heavy contamination. The robust Bayesian methods by using quasi-posterior distributions based on divergences have been also proposed in recent years. In the objective Bayesian framework, the selection of default prior distributions under such quasi-posterior distributions is an important problem. In this study, we provide some properties of reference and moment matching priors under the quasi-posterior distribution based on the -divergence. In particular, we show that the proposed priors are approximately robust under the condition on the contamination distribution without assuming any conditions on the contamination ratio. Some simulation studies are also presented.

A doubly accelerated degradation model based on the inverse Gaussian process and its objective Bayesian analysis

ABSTRACT The accelerated degradation test (ADT) is an effective method for evaluating the lifetime of high-reliability products. In this paper, a doubly accelerated degradation model based on the inverse Gaussian process is proposed to characterize the ADT data, and then an objective Bayesian approach is presented to analyze the model. Some important noninformative priors including the Jeffreys prior and reference priors under different group orderings are derived. The propriety of the posterior distributions under each prior is validated. A simulation study is carried out to show the superiority of objective Bayesian approach compared with the parametric Bootstrap method. Finally, the approach is applied to analyze a carbon film data.

Reference priors via [formula omitted]-divergence for a certain non-regular model in the presence of a nuisance parameter

This paper presents reference priors for non-regular model whose support depends on an unknown parameter. A multi-parameter family which includes both regular and non-regular structures is considered. The resulting priors are obtained by asymptotically maximizing the expected α-divergence between the prior and the corresponding posterior distribution. Some examples of reference priors for typical multi-parameter non-regular distributions such as the location-scale family of distributions and the truncated Weibull distribution are also given.

Objective bayesian inference for quantile ratios in normal models

In medical research, it is important to compare quantiles of certain measures obtained from treatment and control groups, with the quantile ratio showing the effect of the treatment. In particular, inference of the quantile ratio based on large sample methods can be studied using a normal model. In this paper, we develop noninformative priors such as probability matching priors and reference priors for quantile ratios in normal models. It has been proved that the one-at-a-time reference prior satisfies a first-order matching criterion, while the Jeffreys’ and two-group reference priors do not when the variances are equal. Through simulation study and an example based on real data, we also confirm that the proposed probability matching priors match the target coverage probabilities in a frequentist sense even when the sample size is small.

Bias Correction in Clustered Underreported Data

Data quality from poor and socially deprived regions have given rise to many statistical challenges. One of them is the underreporting of vital events leading to biased estimates for the associated risks. To deal with underreported count data, models based on compound Poisson distributions have been commonly assumed. To be identifiable, such models usually require extra and strong information about the probability of reporting the event in all areas of interest, which is not always available. We introduce a novel approach for the compound Poisson model assuming that the areas are clustered according to their data quality. We leverage these clusters to create a hierarchical structure in which the reporting probabilities decrease as we move from the best group to the worst ones. We obtain constraints for model identifiability and prove that only prior information about the reporting probability in areas experiencing the best data quality is required. Several approaches to model the uncertainty about the reporting probabilities are presented, including reference priors. Different features regarding the proposed methodology are studied through simulation. We apply our model to map the early neonatal mortality risks in Minas Gerais, a Brazilian state that presents heterogeneous characteristics and a relevant socio-economical inequality.

Objective Bayesian Prediction of Future Record Statistics Based on the Exponentiated Gumbel Distribution: Comparison with Time-Series Prediction

The interest in the study of record statistics has been increasing in recent years in the context of predicting stock markets and addressing global warming and climate change problems such as cyclones and floods. However, because record values are mostly rare observed, its probability distribution may be skewed or asymmetric. In this case, the Bayesian approach with a reasonable choice of the prior distribution can be a good alternative. This paper presents an objective Bayesian method for predicting future record values when observed record values have a two-parameter exponentiated Gumbel distribution with the scale and shape parameters. For objective Bayesian analysis, objective priors such as the Jeffreys and reference priors are first derived from the Fisher information matrix for the scale and shape parameters, and an analysis of the resulting posterior distribution is then performed to examine its properness and validity. In addition, under the derived objective prior distributions, a simple algorithm using a pivotal quantity is proposed to predict future record values. To validate the proposed approach, it was applied to a real dataset. For a closer examination and demonstration of the superiority of the proposed predictive method, it was compared to time-series models such as the autoregressive integrated moving average and dynamic linear model in an analysis of real data that can be observed from an infinite time series comprising independent sample values.

Bayesian analysis of the covariance matrix of a multivariate normal distribution with a new class of priors

Bayesian analysis for the covariance matrix of a multivariate normal distribution has received a lot of attention in the last two decades. In this paper, we propose a new class of priors for the covariance matrix, including both inverse Wishart and reference priors as special cases. The main motivation for the new class is to have available priors—both subjective and objective—that do not “force eigenvalues apart,” which is a criticism of inverse Wishart and Jeffreys priors. Extensive comparison of these “shrinkage priors” with inverse Wishart and Jeffreys priors is undertaken, with the new priors seeming to have considerably better performance. A number of curious facts about the new priors are also observed, such as that the posterior distribution will be proper with just three vector observations from the multivariate normal distribution—regardless of the dimension of the covariance matrix—and that useful inference about features of the covariance matrix can be possible. Finally, a new MCMC algorithm is developed for this class of priors and is shown to be computationally effective for matrices of up to 100 dimensions.

Bayesian confidence intervals for the difference between variances of delta-lognormal distributions.

Unnatural rainfall fluctuation can result in such severe natural phenomena as drought and floods. This variability not only occurs in areas with unusual natural features such as land formations and drainage but can also be due to human intervention. Since rainfall data often contain zero values, evaluating rainfall change is an important undertaking, which can be estimated via the confidence intervals for the difference between delta-lognormal variances using the highest posterior density-based reference (HPD-ref) and probability-matching (HPD-pm) priors. Simulation results indicate that HPD-pm performances were better than other methods in terms of coverage rates and relative average lengths for the difference in delta-lognormal variances, even with a large difference in variances. To illustrate the efficacy of our proposed methods, we applied them to daily rainfall data sets for the lower and upper regions of northern Thailand.