Non-informative Prior Distributions Research Articles

Bayesian earthquake forecasting approach based on the epidemic type aftershock sequence model

The epidemic type aftershock sequence (ETAS) model is used as a baseline model both for earthquake clustering and earthquake prediction. In most forecast experiments, the ETAS parameters are estimated based on a short and local catalog, therefore the model parameter optimization carried out by means of a maximum likelihood estimation may be not as robust as expected. We use Bayesian forecast techniques to solve this problem, where non-informative flat prior distributions of the parameters is adopted to perform forecast experiments on 3 mainshocks occurred in Southern California. A Metropolis–Hastings algorithm is employed to sample the model parameters and earthquake events. We also show, through forecast experiments, how the Bayesian inference allows to obtain a probabilistic forecast, differently from one obtained via MLE.Graphical

Bayesian Multivariate Logistic Regression for Superiority and Inferiority Decision-Making under Observable Treatment Heterogeneity

The effects of treatments may differ between persons with different characteristics. Addressing such treatment heterogeneity is crucial to investigate whether patients with specific characteristics are likely to benefit from a new treatment. The current paper presents a novel Bayesian method for superiority decision-making in the context of randomized controlled trials with multivariate binary responses and heterogeneous treatment effects. The framework is based on three elements: a) Bayesian multivariate logistic regression analysis with a Pólya-Gamma expansion; b) a transformation procedure to transfer obtained regression coefficients to a more intuitive multivariate probability scale (i.e., success probabilities and the differences between them); and c) a compatible decision procedure for treatment comparison with prespecified decision error rates. Procedures for a priori sample size estimation under a non-informative prior distribution are included. A numerical evaluation demonstrated that decisions based on a priori sample size estimation resulted in anticipated error rates among the trial population as well as subpopulations. Further, average and conditional treatment effect parameters could be estimated unbiasedly when the sample was large enough. Illustration with the International Stroke Trial dataset revealed a trend toward heterogeneous effects among stroke patients: Something that would have remained undetected when analyses were limited to average treatment effects.

Comparative efficacy of energy-restricted dietary interventions in overweight and obese populations: A systematic review and network meta-analysis.

This meta-analysis compared the effectiveness of different energy-restricted diets on body composition, glucose metabolism, and lipid metabolism in overweight and obese populations. Five databases were searched to identify relevant studies in English from inception until July 20, 2023, for randomized controlled trials of at least 2 weeks duration assessing the effects of continuous energy-restricted diets compared with any intermittent energy-restricted diet in obesity adults. The risk of bias was assessed using the Cochrane Risk of Bias Tool version 2.0, while the Grading of Recommendations, Assessment, Development, and Evaluations (GRADE) system was used to assess the certainty of the evidence. A non-informative prior distribution Bayesian network meta-analysis was conducted. Thirty-eight studies (3039 participants) assessing four energy-restricted diets were included. Three RCTs were at high risk of bias with a very low to moderate certainty of evidence. Combined with pairwise comparisons and surface under the cumulative ranking curve, alternate-day fasting may be the best energy restriction regimen with the potential to have the most beneficial effects on various aspects of the obesity population. More rigorously designed and long-term follow-up studies are warranted.

MetaNorm: incorporating meta-analytic priors into normalization of NanoString nCounter data.

Non-informative or diffuse prior distributions are widely employed in Bayesian data analysis to maintain objectivity. However, when meaningful prior information exists and can be identified, using an informative prior distribution to accurately reflect current knowledge may lead to superior outcomes and great efficiency. We propose MetaNorm, a Bayesian algorithm for normalizing NanoString nCounter gene expression data. MetaNorm is based on RCRnorm, a powerful method designed under an integrated series of hierarchical models that allow various sources of error to be explained by different types of probes in the nCounter system. However, a lack of accurate prior information, weak computational efficiency, and instability of estimates that sometimes occur weakens the approach despite its impressive performance. MetaNorm employs priors carefully constructed from a rigorous meta-analysis to leverage information from large public data. Combined with additional algorithmic enhancements, MetaNorm improves RCRnorm by yielding more stable estimation of normalized values, better convergence diagnostics and superior computational efficiency. R Code for replicating the meta-analysis and the normalization function can be found at github.com/jbarth216/MetaNorm.

Estimation of a Shape Parameter of a Gompertz-lindley Distribution Using Bayesian and Maximum Likelihood Methods

The Gompertz-Lindley distribution is an extension of the Lindley distribution with three parameters. It was found to be more flexible for modeling real life events. The distribution contains two shape parameters and a scale parameter. Despite the necessity of parameter estimation theory in modeling, it has not been shown that a method of estimation method is better for any of these three parameters of the Gompertz-Lindley distribution. This paper identifies the best estimation method for the shape parameter of the Gompertz-Lindley distribution by deriving Bayesian estimators for the shape parameter of the distribution using two non-informative prior distributions (Uniform and Jeffery) and an informative prior (gamma) under squared error loss function (SELF), quadratic loss function (QLF) and precautionary loss function (PLF). These estimators were evaluated and the results compared with the maximum likelihood estimation method using Monte Carlo simulations with the mean square error (MSE) as a criterion for choosing the best estimator.

Bayesian approach for a 2 x 2 crossover design with repeated measures: a simulation study

In crossover designs, the subjects receive all treatments, according to the groups of sequences formed. Therefore, if carryover effects are present in the model, inferences about the treatments effects become difficult. Furthermore, repeated measures of the response variable can be taken over time in the same experimental unit; however, these measures may be correlated. In this way, we aimed to analyze a 2 x 2 crossover design with repeated measures within the treatment period, using a Bayesian approach. A simulation study was performed to evaluate the performance. The posterior estimates of the model parameters were obtained under non-informative prior distributions and the normal likelihood function. The model performed well with a sample size of 20 subjects, showing even better results with samples of 100 subjects. With larger samples, exact tests for the differences in carryover effects and time effects were obtained. However, the test of time effect proved to be powerful even with small samples. In turn, considering carryover effects different from zero did not influence the estimates of treatment differences, although biased estimates of the period effect were obtained.

Reducing the false discovery rate of preclinical animal research with Bayesian statistical decision criteria.

The success of preclinical research hinges on exploratory and confirmatory animal studies. Traditional null hypothesis significance testing is a common approach to eliminate the chaff from a collection of drugs, so that only the most promising treatments are funneled through to clinical research phases. Balancing the number of false discoveries and false omissions is an important aspect to consider during this process. In this paper, we compare several preclinical research pipelines, either based on null hypothesis significance testing or based on Bayesian statistical decision criteria. We build on a recently published large-scale meta-analysis of reported effect sizes in preclinical animal research and elicit a non-informative prior distribution under which both approaches are compared. After correcting for publication bias and shrinkage of effect sizes in replication studies, simulations show that (i) a shift towards statistical approaches which explicitly incorporate the minimum clinically important difference reduces the false discovery rate of frequentist approaches and (ii) a shift towards Bayesian statistical decision criteria can improve the reliability of preclinical animal research by reducing the number of false-positive findings. It is shown that these benefits hold while keeping the number of experimental units low which are required for a confirmatory follow-up study. Results show that Bayesian statistical decision criteria can help in improving the reliability of preclinical animal research and should be considered more frequently in practice.

Particle MCMC in Forecasting Frailty-Correlated Default Models with Expert Opinion

Predicting corporate default risk has long been a crucial topic in the finance field, as bankruptcies impose enormous costs on market participants as well as the economy as a whole. This paper aims to forecast frailty-correlated default models with subjective judgements on a sample of U.S. public non-financial firms spanning January 1980–June 2019. We consider a reduced-form model and adopt a Bayesian approach coupled with the Particle Markov Chain Monte Carlo (Particle MCMC) algorithm to scrutinize this problem. The findings show that the 1-year prediction for frailty-correlated default models with different prior distributions is relatively good, whereas the prediction accuracy ratios for frailty-correlated default models with non-informative and subjective prior distributions over various prediction horizons are not significantly different.

Discussion of “Specifying prior distributions in reliability applications”

Discussion of “Specifying prior distributions in reliability applications”

Geometric properties of noninformative priors based on the chi-square divergence

Recently, a noninformative prior distribution that is different from the Jeffreys prior was derived as an extension of Bernardo's reference prior based on the chi-square divergence. We summarize this result in terms of information geometry and clarify some geometric properties. Specifically, we show that it corresponds to a parallel volume element and can be written as a power of the Jeffreys prior in flat model manifolds.

Bayesian Log-Rank Test

Comparison of two survival curves is a fundamental problem in survival analysis. Although abundant frequentist methods have been developed for comparing survival functions, inference procedures from the Bayesian perspective are rather limited. In this article, we extract the quantity of interest from the classic log-rank test and propose its Bayesian counterpart. Monte Carlo methods, including a Gibbs sampler and a sequential importance sampling procedure, are developed to draw posterior samples of survival functions and a decision rule of hypothesis testing is constructed for making inference. Via simulations and real data analysis, the proposed Bayesian log-rank test is shown to be asymptotically equivalent to the classic one when noninformative prior distributions are used, which provides a Bayesian interpretation of the log-rank test. When using the correct prior information from historical data, the Bayesian log-rank test is shown to outperform the classic one in terms of power. R codes to implement the Bayesian log-rank test are also provided with step-by-step instructions.

Updating knowledge in estimating the genetics parameters: Multi-trait and Multi-Environment Bayesian analysis in rice

Among the multi-trait models selected to study several traits and environments jointly, the Bayesian framework has been a preferred tool when constructing a more complex and biologically realistic model. In most cases, non-informative prior distributions are adopted in studies using the Bayesian approach. However, the Bayesian approach presents more accurate estimates when informative prior distributions are used. The present study was developed to evaluate the efficiency and applicability of multi-trait multi-environment (MTME) models within a Bayesian framework utilizing a strategy for eliciting informative prior distribution using previous data on rice. The study involved data pertaining to rice (Oryza sativa L.) genotypes in three environments and five crop seasons (2010/2011 until 2014/2015) for the following traits: grain yield (GY), flowering in days (FLOR) and plant height (PH). Variance components, genetic and non-genetic parameters were estimated using the Bayesian method. In general, the informative prior distribution in Bayesian MTME models provided higher estimates of individual narrow-sense heritability and variance components, as well as minor lengths for the highest probability density interval (HPD), compared to their respective non-informative prior distribution analyses. More informative prior distributions make it possible to detect genetic correlations between traits, which cannot be achieved with non-informative prior distributions. Therefore, this mechanism presented to update knowledge for an elicitation of an informative prior distribution can be efficiently applied in rice breeding programs.

Bayesian D-Optimal Design of Experiments with Quantitative and Qualitative Responses

Systems with both quantitative and qualitative responses are widely encountered in many applications. Design of experiment methods are needed when experiments are conducted to study such systems. Classic experimental design methods are unsuitable here because they often focus on one type of response. In this paper, we develop a Bayesian D-optimal design method for experiments with one continuous and one binary response. Both noninformative and conjugate informative prior distributions on the unknown parameters are considered. The proposed design criterion has meaningful interpretations regarding the D-optimality for the models for both types of responses. An efficient point-exchange search algorithm is developed to construct the local D-optimal designs for given parameter values. Global D-optimal designs are obtained by accumulating the frequencies of the design points in local D-optimal designs, where the parameters are sampled from the prior distributions. The performances of the proposed methods are evaluated through two examples.

Bayesian Estimation of a Scale Parameter of the Gumbel-Lomax Distribution Using Informative and Non Informative Priors

Estimating the scale parameter of the Gumbel-Lomax Distribution using the Bayesian method of estimation and evaluating the estimators by assuming two non-informative prior distributions and one informative prior distribution is very important for the general application of the Gumbel-Lomax distribution. These estimators are obtained using the squared error loss function (SELF), Quadratic loss function (QLF) and precautionary loss function (PLF). The posterior distributions of the scale parameter of the Gumbel-Lomax distribution are derived and the Estimators are also obtained using the above mentioned priors and loss functions. Furthermore, a simulation using a package in R software is carried out to assess the performance of the estimators by making use of the Mean Squared Errors of the Estimators under the Bayesian approach and Maximum likelihood method. Our results show that Bayesian Method using PLF under all priors produces the best estimators of the scale parameter compared to estimators using the Maximum Likelihood method, SELF and QLF under all the priors irrespective of the values of the parameters and the different sample sizes. It is also discovered that the other parameters have no effect on the estimators of the scale parameter.

Statistical Inference of Stress-Strength Reliability of Gompertz Distribution under Type II Censoring

This paper develops the problem of estimating stress-strength reliability for Gompertz lifetime distribution. First, the maximum likelihood estimation (MLE) and exact and asymptotic confidence intervals for stress-strength reliability are obtained. Then, Bayes estimators under informative and noninformative prior distributions are obtained by using Lindley approximation, Monte Carlo integration, and MCMC. Bayesian credible intervals are constructed under these prior distributions. Also, simulation studies are used to illustrate these inference methods. Finally, a real dataset is analyzed to show the implementation of the proposed methodologies.

Inference for a general family of inverted exponentiated distributions under unified hybrid censoring with partially observed competing risks data

In this paper, statistical inference of a competing risks model is developed based on a unified hybrid censoring scheme when the latent failure times follow a general family of inverted exponentiated distributions, which covers a wide range of lifetime distributions. Point and interval estimation methods for estimating the model parameters based on maximum likelihood and Bayesian approaches under non-restricted and restricted parameter spaces are developed. The Bayes estimates are obtained under the squared error loss function with non-informative and informative prior distributions. The existence and uniqueness of the maximum likelihood estimates of the model parameters are proved. A Monte Carlo simulation study is carried out to evaluate the performance of the proposed estimation procedures. To illustrate the proposed inferential procedures, a real data analysis is provided. Finally, some concluding remarks and future research directions are presented.

On the Application of Probabilistic Decline Curve Analysis to Unconventional Reservoirs

Summary Several authors have worked on combining decline curve analysis (DCA) models and stochastic algorithms for probabilistic DCAs. However, there are no publications on the application of these probabilistic decline curve models to all the major shale basins in the United States. Also, several empirical and analytical decline curve models have been developed to fit historical production data better; there is no systematic investigation of the relevance of the efforts on new model development compared with the efforts to quantify the uncertainty associated with the “noise” in the historical data. This work compares the uncertainty associated with determining the best-fit model (epistemic uncertainty) with the uncertainty associated with the historical data (aleatoric uncertainty) and presents a procedure to find DCA-stochastic algorithm combinations that encompass the epistemic uncertainty. We investigated two Bayesian methods—the approximate Bayesian computation and the Gibbs sampler—and two frequentist methods—the conventional bootstrap (BS) and modified BS (MBS). These stochastic algorithms were combined with five empirical DCA models (Arps, Duong, power law, logistic growth, and stretched exponential decline) and the analytical Jacobi theta-2 model. We analyzed historical production data from 1,800 wells (300 wells from each of the six major shale basins studied) with historical data lengths ranging from 12 to 60 months. We show the errors associated with the assumption of a uniform distribution for the model parameters and present an approach for integrating informative prior (IP) probabilistic distributions instead of the noninformative prior (NIP) or uniform prior distributions. Our results indicate the superior performance of the Bayesian methods, especially at short hindcasts (12–24 months of production history). We observed that the duration of the historical production data was the most critical factor. Using long hindcasts (up to 60 months) leveled the performance of all probabilistic methods regardless of the decline curve model or statistical methodology used. Additionally, we showed that it is possible to find DCA-stochastic model combinations that reflect the epistemic uncertainty in most of the shale basins investigated. The novelty of this work lies in the development of IPs for the Bayesian methodologies and the development of a systematic approach to determine the combination of statistical methods and DCA models that encompasses the epistemic uncertainty. The proposed approach was implemented using open-source software packages to make our results reproducible and to facilitate its practical application in forecasting production in unconventional oil and gas reservoirs.

A New Process Performance Index for the Weibull Distribution with a Type-I Hybrid Censoring Scheme

A new life performance index is proposed for evaluating the quality of lifetime products. The maximum likelihood estimation method and the Bayesian approaches using informative and non-informative prior distributions are utilized to infer the parameters of the Weibull distribution and the proposed new life performance index under a Type-I hybrid censoring scheme. Monte Carlo simulation results show that two Bayesian approaches outperform the maximum likelihood estimation method in terms of the measures of relative bias, relative mean square error, and coverage probability for the point and confidence interval estimators, respectively. The Bayesian approach using a non-informative prior distribution is recommended if the knowledge of setting up the hyper-parameters in the informative prior distribution is not available. Two real data sets are provided for illustration.

Designing efficient Bayesian sampling plans based on the simple step-stress test under random stress-change time for censored data

ABSTRACT This paper studies the problem about how to design an efficient Bayesian sampling plan based on a simple step-stress test with a random stress-change time for censored data. For brevity, the step-stress test with random stress-change time is briefly called SSRSCT. A Bayesian sampling plan (BSP) through the SSRSCT is called BSPA. First, we derive the BSPA with data based on Type-II censoring for a general loss function. Given gamma and noninformative uniform prior distributions, an explicit expression of the Bayes decision function under a certain loss function is derived. By applying a curtailment procedure to the preceding Bayes decision function, a new Bayesian sampling plan, called an efficient Bayesian sampling plan (EBSPA), is constructed. It is shown that the Bayes risk of EBSPA is less than or equal to that of BSPA. This indicates that EBSPA is more efficient than the BSPA. Comparisons among some BSPAs and the proposed EBSPA are given. Monte Carlo simulations are carried out, and results indicate that the EBSPA outperforms those BSPAs. The EBSPA is recommended.

Bayesian Approach for Confidence Intervals of Variance on the Normal Distribution

This research aims to compare estimating the confidence intervals of variance based on the normal distribution with the primary method and the Bayesian approach. The maximum likelihood is the well-known method to approximate variance, and the Chi-squared distribution performs the confidence interval. The central Bayesian approach forms the posterior distribution that makes the variance estimator, which depends on the probability and prior distributions. Most introductory prior information looks for the availability of the prior distribution, informative prior distribution, and noninformative prior distribution. The gamma, Chi-squared, and exponential distributions are defined in the prior distribution. The informative prior distribution uses the Markov Chain Monte Carlo (MCMC) method to draw the random sample from the posterior distribution. The Fisher information performs the Wald confidence interval as the noninformative prior distribution. The interval estimation of the Bayesian approach is obtained from the central limit theorem. The performance of these methods considers the coverage probability and minimum value of the average width. The Monte Carlo process simulates the data from a normal distribution with the true parameter of mean and several variances and the sample sizes. The R program generates the simulated data repeated 10,000 times in each situation. The results showed that the maximum likelihood method employed on the small sample sizes. The best confidence interval estimation was when sample sizes increased the Bayesian approach with an available prior distribution. Overall, the Wald confidence interval tended to outperform the large sample sizes. For application in real data, we expressed the reported airborne particulate matter of 2.5 in Bangkok, Thailand. We used the 10–1000 records to estimate the confidence interval of variance and evaluated the interval width. The results are similar to those of the simulation study.