Bayesian Prediction Intervals Research Articles

Shifted Exponential Distribution: Bayesian Estimation, Prediction and Expected Test Time under Progressive Censoring

Abstract In this article, we present Bayesian estimation of a shifted exponential distribution assuming progressive type-II censoring with random removals. Bayes estimators and their respective posterior risks are derived under the squared error loss function and the precautionary loss function. Moreover, the credible intervals and expected experiment time along with the Bayesian predictive intervals for one- and two-sample cases are also discussed. A comparison of the expected experiment time under complete and type-II progressive censoring is also a part of this study. Finally, a real-life example is provided to illustrate the proposed methodology.

Consequences of Lack of Parameterization Invariance of Non-informative Bayesian Analysis for Wildlife Management: Survival of San Joaquin Kit Fox and Declines in Amphibian Populations

Computational convenience has led to widespread use of Bayesian inference with vague or flat priors to analyze statistical models in ecology. Vague priors are claimed to be objective and to `let the data speak'. However, statisticians have long disputed these claims and have criticized the use of vague priors from philosophical to computational to pragmatic reasons. One of the major criticisms is that the inferences based on non-informative priors are generally dependent on the parameterization of the models. Ecologists, unfortunately, often dismiss such criticisms as having no practical implications. One argument is that for large sample sizes, the priors do not matter. The problem with this argument is that, in practice, one does not know whether or not the observed sample size is sufficiently large for the effect of the prior to vanish. It intricately depends on the complexity of the model and the strength of the prior. We study the consequences of parameterization dependence of the non-informative Bayesian analysis in the context of population viability analysis and occupancy models and at the commonly obtained sample sizes. We show that they can have significant impact on the analysis, in particular on prediction, and can lead to strikingly different managerial decisions. In general terms, the consequences are: (1) All subjective Bayesian inferences can be masqueraded as objective (flat prior) Bayesian inferences, (2) Induced priors on functions of parameters are not flat, thus leading to cryptic biases in scientific inferences, (3) Unrealistic independent priors for multiparameter models lead to unrealistic priors on induced parameters, (4) Bayesian prediction intervals may not have correct coverage, thus leading to errors in decision making, (5) Reparameterization to facilitate MCMC convergence may influence scientific inference. Given the wide spread applicability of the hierarchical models and uncritical use of non-informative Bayesian analysis in ecology, researchers should be cautious about using vague priors as a default choice in practical situations.

Patent Data Analysis of Artificial Intelligence Using Bayesian Interval Estimation

Technology analysis is one of the important tasks in technology and industrial management. Much information about technology is contained in the patent documents. So, patent data analysis is required for technology analysis. The existing patent analyses relied on the quantitative analysis of the collected patent documents. However, in the technology analysis, expert prior knowledge should also be considered. In this paper, we study the patent analysis method using Bayesian inference which considers prior experience of experts and likelihood function of patent data at the same time. For keyword data analysis, we use Bayesian predictive interval estimation with count data distributions such as Poisson. Using the proposed models, we forecast the future trends of technological keywords of artificial intelligence (AI) in order to know the future technology of AI. We perform a case study to provide how the proposed method can be applied to real areas. In this paper, we retrieve the patent documents related to AI technology, and analyze them to find the technological trend of AI. From the results of AI technology case study, we can find which technological keywords are more important or critical in the entire structure of AI industry. The existing methods for patent keyword analysis were depended on the collected patent documents at present. But, in technology analysis, the prior knowledge by domain experts is as important as the collected patent documents. So, we propose a method based on Bayesian inference for technology analysis using the patent documents. Our method considers the patent data analysis with the prior knowledge from domain experts.

The temperature and light responses on the photosynthesis of two freshwater red algae, Virescentia helminthosa and Sheathia arcuata (Batrachospermaceae), from Japan

The temperature and light responses of photosynthesis in two freshwater red algae, Virescentia helminthosa and Sheathia arcuata (Batrachospermaceae), were determined by a pulse amplitude modulation (PAM) chlorophyll fluorometer and dissolved oxygen sensors. Net oxygenic photosynthesis–photosynthetically active radiation (PAR) models of V. helminthosa and S. arcuata revealed similar low PAR-adapted responses, with a compensation PAR (Ec) of 6.95 and 11.5 μmol photons m−2 s−1 (5.58–8.42 and 9.10–11.42, 95% Bayesian prediction interval, BPI) and saturating PAR (Ek) of 18.8 and 17.7 μmol photons m−2 s−1 (14.5–24.7 and 13.0–23.90, BPI), respectively. A temperature-dependent model of net photosynthesis and dark respiration for two species also showed similar temperature responses, and the gross photosynthetic rate (GPmax), 1.79 and 1.19 μg O2 gww−1 min−1 (1.62–1.96 and 1.08–1.29, BPI), was highest at 26.4 and 30.3 °C (23.9–28.7 and 28.3–32.1, BPI). The maximum quantum yields (Fv/Fm) for two species also had similar responses with respect to temperature; however, it was generally stable at low temperatures (8–20 °C) with the highest value of 0.52 and 0.56 (0.49–0.54 and 0.54–0.58, BPI) occurring at 18.5 and 20.9 °C (17.1–19.7 and 19.8–21.9, BPI). Continuous exposure (12 h) to PAR of 100 (low) and 1000 (high) μmol photons m−2 s−1 at 12, 16, and 24 °C revealed greater declines in their effective quantum yield (ΦPSII) in the two species under high PAR. Nevertheless, their Fv/Fm mostly recovered after a subsequent 12-h dim-light acclimation, suggesting the potential of recovery from daytime chronic photoinhibition.

Estimation and prediction based on type-I hybrid censored data from the Poisson-Exponential distribution

This paper considers the problems of estimation and prediction when lifetime data following Poisson-exponential distribution are observed under type-I hybrid censoring. For both the problems, we compute point and associated interval estimates under classical and Bayesian approaches. For point estimates in the problem of estimation, we compute maximum likelihood estimates using Newton–Raphson, Expectation-Maximization and Stochastic Expectation-Maximization algorithms under classical approach, and under Bayesian approach we compute Bayes estimates with the help of Lindley and importance sampling technique under informative and non-informative priors using symmetric and asymmetric loss functions. The associated interval estimates are obtained using the Fisher information matrix and Chen and Shao method respectively under classical and Bayesian approaches. Further, the predictive point estimates and associated predictive interval estimates are computed by making use of best unbiased and conditional median predictors under classical approach, and Bayesian predictive and associated Bayesian predictive interval estimates in the problem of prediction. We analysis real data set, and conduct Monte Carlo simulation study for the comparison of various proposed methods of estimation and prediction. Finally, a conclusion is given.

Photosynthetic activity in two heteromorphic life‐history stages of a freshwater red alga, Thorea gaudichaudii (Thoreales) from Japan, in response to an irradiance and temperature gradient

SUMMARYWe determined the effect of irradiance and temperature on the photosynthesis of two heteromorphic life‐history stages of an endangered freshwater red alga, Thorea gaudichaudii (Thoreales) by laboratory and field measurements. Net oxygenic photosynthesis–irradiance models of macroscopic and microscopic life‐history stages revealed similar low irradiance‐adapted responses, with a compensation irradiance (Ec) of 6.71 and 2.56 μmol photons m−2 s−1 (4.30–9.13 and 0.13–7.19, 95% Bayesian prediction interval, BPI) and saturating irradiance (Ek) of 26.6 and 30.0 μmol photons m−2 s−1 (19.0–37.4 and 12.1–63.0, BPI), respectively. A temperature‐dependent model of net photosynthesis and dark respiration in macroscopic and microscopic stages also showed similar temperature responses, and the gross photosynthetic rate (GPmax), 3.54 and 6.34 μg O2 gww−1 min−1 (3.10–3.99 and 5.31–8.21, BPI), was highest at 32.1 and 35.7°C (29.8–34.0 and 29.5–48.6, BPI). The maximum quantum yields (F v/F m) in macroscopic and microscopic stages were also similar in response with respect to temperature; however, it was somewhat steady at low temperatures with the highest value of 0.54 and 0.62 (0.54–0.55 and 0.61–0.63, BPI) at 17.8 and 15.0°C (16.7–18.8 and 12.3–17.1, BPI). The effective quantum yield (Φ PSII) in macroscopic and microscopic stages was also negatively correlated with irradiance, which decreased after 12 h of continuous exposure to 50 (low) and 1000 (high) μmol photons m−2 s−1 at 12 and 22°C. Large declines of Φ PSII and subsequent failure of F v/F m recovery were particularly enhanced at high irradiance, signifying photoinhibition. Diurnal change of Φ PSII and incident irradiance of the macroscopic stage under the field measurement revealed the midday depression of Φ PSII; however, there was little direct sunlight due to shading by the trees, and algae were occurring in the shaded locations in the freshwater spring.

Bayesian simultaneous prediction intervals and bounds for a finite population

Key PointSimultaneous prediction intervals and bounds provide a statistical characterization of a proportion of a finite population. This article considers these predictions from a Bayesian inferential approach and different ways to evaluate them including simulation.

Statistical inference based on generalized Lindley record values

This paper addresses the problems of frequentist and Bayesian estimation for the unknown parameters of generalized Lindley distribution based on lower record values. We first derive the exact explicit expressions for the single and product moments of lower record values, and then use these results to compute the means, variances and covariance between two lower record values. We next obtain the maximum likelihood estimators and associated asymptotic confidence intervals. Furthermore, we obtain Bayes estimators under the assumption of gamma priors on both the shape and the scale parameters of the generalized Lindley distribution, and associated the highest posterior density interval estimates. The Bayesian estimation is studied with respect to both symmetric (squared error) and asymmetric (linear-exponential (LINEX)) loss functions. Finally, we compute Bayesian predictive estimates and predictive interval estimates for the future record values. To illustrate the findings, one real data set is analyzed, and Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and prediction.

Bayesian prediction of future observations from weighted exponential distribution constant-stress model based on Type-II hybrid censored data

In this paper, the problem of Bayesian prediction intervals for a future observations from weighted exponential distribution is concerned. Constant-stress partially accelerated life test under Type-II hybrid censoring scheme of the observed data is used. One- and two-sample Bayesian prediction intervals for a future observations based on Type-II hybrid censored data are derived. Markov Chain Monte Carlo (MCMC) technique is used to find Bayesian predictive intervals because one- and two-sample Bayesian predictive survival function cannot be obtained in closed-form. Finally, some numerical results are presented to illustrate all the inferential results developed in this paper.

Bayesian estimation of the mixture of Burr Type-XII distributions using doubly censored data

This study discusses the Bayesian and maximum likelihood estimation methods for analyzing the data from a 3-component mixture of Burr Type-XII probability distributions. The maximum likelihood estimators with their variances cannot be obtained in an explicit form and thus an iterative procedure is used to calculate them numerically. Contrary to this, elegant closed form algebraic expressions of Bayes estimators and their posterior risks are derived. Using the informative and noninformative priors, the posterior predictive distributions along with predictive intervals are also discussed. A method of eliciting hyperparameters using prior predictive distribution is also a part of this study. Some interesting properties (including, posterior risks and Bayesian predictive intervals) of Bayes estimators and their behavior across different sample sizes, left and right test termination times, informative prior versus Jeffreys noninformative prior, are provided via a detailed Monte Carlo simulation study. To assess the suitability and application of the proposed model, a real life data example is also discussed in this article. Based on the simulated results and real data application, it is concluded that the IP paired with DLF (SELF) is a more suitable for estimating mixing component.

Bayesian analysis for dependent competing risks model with masked causes of failure in step-stress accelerated life test under progressive hybrid censoring

This article considers the statistical analysis of dependent competing risks model with masked causes of failure in step-stress accelerated life test, where the failure times of the competing risks at each stress level follow Weibull distribution and they are related to the Khamis–Higgins model assumptions. In the usual step-stress experiment, the expected lifetime of the experimental unit is shortened as the stress level increases. We provided Bayesian inference of the unknown parameters under this restriction using the prior assumption. In addition, Bayesian predictive estimates and prediction intervals for the future observations are obtained. Finally, simulations are performed to demonstrate the performances of the estimates, and one data set has been analyzed for illustrative purposes.

Effect of temperature and PAR on photosynthesis of an endangered freshwater red alga, Thorea okadae, from Kagoshima, Japan

Abstract The effect of photosynthetically active radiation (PAR) and temperature on the photosynthesis of an endangered Japanese freshwater red alga, Thorea okadae (Thoreaceae, Thoreales), was determined by using pulse-amplitude modulation (PAM) fluorometry and dissolved oxygen sensors. The macroscopic life history stage (gametophyte) was observed on pebbles and cobbles in a shallow riverbed from early winter to late spring (December–May), in direct sunlight (∼800 μmol photons m−2 s−1) during winter noon time with a clear sky. A net oxygenic photosynthesis–PAR model revealed that the net photosynthetic rate increased linearly and saturated, with a compensation (Ec) and saturating PAR (Ek) of 13.3 μmol photon m−2 s−1 (10.3–16.3, 95% Bayesian prediction interval, BPI) and 55.2 μmol photon m−2 s−1 (42.2–72.9, 95% BPI), respectively. A temperature-dependent (i.e., 8–36°C range) model of net photosynthesis and dark respiration was fitted and showed that the gross photosynthetic rate, 17.3 μg O2 gww−1 min−1 (16...

On Estimation and Prediction for the Inverted Kumaraswamy Distribution Based on General Progressive Censored Samples

In this article, the problem of estimating unknown parameters of the inverted kumaraswamy (IKum) distribution is considered based on general progressive Type-II censored Data. The maximum likelihood (MLE) estimators of the parameters are obtained while the Bayesian estimates are obtained using the squared error loss(SEL) as symmetric loss function. Also we used asymmetric loss functions as the linear-exponential loss (LINEX), generalized entropy (GE) and Al-Bayyati loss function (AL-Bayyati). Lindely's approximation method is used to evaluate the Bayes estimates. We also derived an approximate confidence interval for the parameters of the inverted Kumaraswamy distribution. Two-sample Bayesian prediction intervals are constructed with an illustrative example. Finally, simulation study concerning different sample sizes and different censoring schemes were reported.

Bayesian Analysis of Heterogeneous Doubly Censored Lifetime Data using the 3-Component Mixture of Rayleigh Distributions: A Monte Carlo Simulation Study

This article is about Bayesian estimation of parameters of a heterogeneous 3-component mixture of Rayleigh distributions (3-CMRD) generating a mixture data. Being the most popular and reasonable sampling scheme in reliability and survival analyses, the doubly censored sampling scheme is considered. The Bayes estimators and their posterior risks are derived under various situations. In addition, elicitation of hyperparameters is presented. Algebraic expressions for posterior predictive distribution and Bayesian predictive intervals are derived. Assuming the informative and the non-informative priors, a comprehensive Monte Carlo simulation is conducted to examine the performance of the Bayes estimators under symmetric and asymmetric loss functions. Finally, to highlight the practical importance, the proposed 3-compnent mixture model is applied to a doubly censored lifetime data from a real life situation. It is observed that the analysis of doubly censored data in Bayesian framework, the SRIGP paired with SELF (DLF) is suitable choice for estimating mixing proportion (component) parameters.

Statistical Inference and Optimum Life Testing Plans Under Type-II Hybrid Censoring Scheme

This article considers estimation of unknown parameters and prediction of future observations of a generalized exponential distribution based on Type-II hybrid censored data. Bayes point and HPD interval estimates of the unknown parameters are obtained under the assumption of independent gamma priors. Different classical and Bayesian point predictors and prediction intervals are obtained in two-sample situation against squared error loss function. The optimum censoring schemes are computed under various optimality criteria. Monte Carlo simulations are performed to compare different methods and two data sets are analyzed for illustrative purposes.

Estimations and Prediction for the Compound Rayleigh Distribution Based on Upper Record Values

This article considers estimation of the unknown parameters for the compound Rayleigh distribution (CRD) based on upper record values. We have derived the maximum likelihood (ML) and Bayesian estimators for the unknown two parameters, as well as the reliability and hazard functions. We obtained Bayes estimators on the basis of the symmetric (squared error) and asymmetric (linear exponential (LINEX) and general entropy (GE)) loss functions. It has been seen that the symmetric and asymmetric Bayes estimators are obtained in closed forms. Furthermore, Bayesian prediction interval of the future upper record values are discussed and obtained. Finally, estimation of the parameters, practical examples of real record values and simulated record values are given to illustrate the theoretical results of prediction interval.

Bayesian prediction intervals for assessing P-value variability in prospective replication studies

Increased availability of data and accessibility of computational tools in recent years have created an unprecedented upsurge of scientific studies driven by statistical analysis. Limitations inherent to statistics impose constraints on the reliability of conclusions drawn from data, so misuse of statistical methods is a growing concern. Hypothesis and significance testing, and the accompanying P-values are being scrutinized as representing the most widely applied and abused practices. One line of critique is that P-values are inherently unfit to fulfill their ostensible role as measures of credibility for scientific hypotheses. It has also been suggested that while P-values may have their role as summary measures of effect, researchers underappreciate the degree of randomness in the P-value. High variability of P-values would suggest that having obtained a small P-value in one study, one is, ne\tvertheless, still likely to obtain a much larger P-value in a similarly powered replication study. Thus, “replicability of P-value” is in itself questionable. To characterize P-value variability, one can use prediction intervals whose endpoints reflect the likely spread of P-values that could have been obtained by a replication study. Unfortunately, the intervals currently in use, the frequentist P-intervals, are based on unrealistic implicit assumptions. Namely, P-intervals are constructed with the assumptions that imply substantial chances of encountering large values of effect size in an observational study, which leads to bias. The long-run frequentist probability provided by P-intervals is similar in interpretation to that of the classical confidence intervals, but the endpoints of any particular interval lack interpretation as probabilistic bounds for the possible spread of future P-values that may have been obtained in replication studies. Along with classical frequentist intervals, there exists a Bayesian viewpoint toward interval construction in which the endpoints of an interval have a meaningful probabilistic interpretation. We propose Bayesian intervals for prediction of P-value variability in prospective replication studies. Contingent upon approximate prior knowledge of the effect size distribution, our proposed Bayesian intervals have endpoints that are directly interpretable as probabilistic bounds for replication P-values, and they are resistant to selection bias. We showcase our approach by its application to P-values reported for five psychiatric disorders by the Psychiatric Genomics Consortium group.

A Naive Bayesian Wind Power Interval Prediction Approach Based on Rough Set Attribute Reduction and Weight Optimization

Intermittency and uncertainty pose great challenges to the large-scale integration of wind power, so research on the probabilistic interval forecasting of wind power is becoming more and more important for power system planning and operation. In this paper, a Naive Bayesian wind power prediction interval model, combining rough set (RS) theory and particle swarm optimization (PSO), is proposed to further improve wind power prediction performance. First, in the designed prediction interval model, the input variables are identified based on attribute significance using rough set theory. Next, the Naive Bayesian Classifier (NBC) is established to obtain the prediction power class. Finally, the upper and lower output weights of NBC are optimized segmentally by PSO, and are used to calculate the upper and lower bounds of the optimal prediction intervals. The superiority of the proposed approach is demonstrated by comparison with a Naive Bayesian model with fixed output weight, and a rough set-Naive Bayesian model with fixed output weight. It is shown that the proposed rough set-Naive Bayesian-particle swarm optimization method has higher coverage of the probabilistic prediction intervals and a narrower average bandwidth under different confidence levels.

Interval Estimation of Latent Variable Scores in Item Response Theory

The uncertainty arising from item parameter estimation is often not negligible and must be accounted for when calculating latent variable (LV) scores in item response theory (IRT). It is particularly so when the calibration sample size is limited and/or the calibration IRT model is complex. In the current work, we treat two-stage IRT scoring as a predictive inference problem: The target of prediction is a random variable that follows the true posterior of the LV conditional on the response pattern being scored. Various Bayesian, fiducial, and frequentist prediction intervals of LV scores, which can be obtained from a simple yet generic Monte Carlo recipe, are evaluated and contrasted via simulations based on several measures of prediction quality. An empirical data example is also presented to illustrate the use of candidate methods.

Statistical Inference of Exponentiated Moment Exponential Distribution Based on Lower Record Values

Based on lower record values, we first derive the exact explicit expressions as well as recurrence relations for the single and product moments of record values and then use these results to compute the means, variances and coefficient of skewness and kurtosis of exponentiated moment exponential distribution (EMED), a new extension of moment exponential distribution, recently introduced by Hasnain (Exponentiated moment exponential distribution. Ph.D. Thesis, 2013). Next we obtain the maximum likelihood estimators of the unknown parameters and the approximate confidence intervals of the EMED. Finally, we consider Bayes estimation under the symmetric and asymmetric loss functions using gamma priors for both shape and scale parameters. We have also derived the Bayes interval of this distribution and discussed both frequentist and the Bayesian prediction intervals of the future record values based on the observed record values. Monte Carlo simulations are performed to compare the performances of the proposed methods, and a data set has been analyzed for illustrative purposes.