Beta Prior Distribution Research Articles

Estimation of Steady State Probability Distribution of System Size in M/M/1 Queue

Recently Choudhury and Borthakur (2008) obtained classical and Bayesian estimators of performance measures based on a random sample of size n from a geometric distribution with mean ρ / (1− ρ) , which is the steady state probability distribution of system size. Here weobtain classical estimators, the Maximum Likelihood (ML) and Uniformly Minimum Variance Unbiased (UMVU) estimators, and Bayesian estimators of Pk , k=0,1,2,.... relative to beta prior distribution and Weighted Squared Error Loss (WSEL) function as well as relative to Standard Two-Sided Power (STSP) distribution and squared error loss / WSEL functions. Bayes, ML and UMVU estimates of the probability that the server is idle are also computed. Also Consistent Asymptotic Normality (CAN) for Pk , k=0,1,2,.... are examined.

The use of Bayesian methods for evaluating the performance of a virus-like particles-based ELISA for serology of hepatitis E virus infection in swine

In the absence of commercial kits for serological studies specifically for hepatitis E virus (HEV) infection of swine, the capsid protein of a swine genotype 3 HEV was expressed using a recombinant baculovirus. This antigen was used in an enzyme-linked immunosorbent assay (ELISA) for detection of anti-HEV antibodies. Estimation of sensitivity and specificity was carried out in comparison with a commercial serological assay using sera from pigs infected experimentally with HEV genotype 3, negative sera from SPF pigs and swine sera collected at slaughterhouses. Since a “gold standard” is not available, a latent-class Bayesian approach for correlated tests was used to estimate the sensitivity and specificity of both tests. External field and experimental data were used to determine the parameters of Beta prior distributions. The estimated mean sensitivity of the commercial test was 0.47 with a 95% credibility interval of 0.39–0.55, whereas the estimated mean sensitivity of the genotype 3-based assay was 0.92 [0.81–0.99]. The estimated specificities were 0.98 [0.93–0.99] and 0.98 [0.95–0.99] for the genotype 3-based test and commercial test, respectively. In conclusion, genotype 3-antigen derived from swine HEV is a better candidate for assessing hepatitis E serology in swine.

Fast Diagnosis With Sensors of Uncertain Quality

This correspondence presents an approach to the detection and isolation of component failures in large-scale systems. In the case of sensors that report at rates of 1 Hz or less, the algorithm can be considered real time. The input is a set of observed test results from multiple sensors, and the algorithm's main task is to deal with sensor errors. The sensors are assumed to be of threshold test (pass/fail) type, but to be vulnerable to noise, in that occasionally true failures are missed, and likewise, there can be false alarms. These errors are further assumed to be independent conditioned on the system's diagnostic state. Their probabilities, of missed detection and of false alarm, are not known a priori and must be estimated (ideally along with the accuracies of these estimates) online, within the inference engine. Further, recognizing a practical concern in most real systems, a sparsely instantiated observation vector must not be a problem. The key ingredients to our solution include the multiple-hypothesis tracking philosophy to complexity management, a Beta prior distribution on the sensor errors, and a quickest detection overlay to detect changes in these error rates when the prior is violated. We provide results illustrating performance in terms of both computational needs and error rate, and show its application both as a filter (i.e., used to "clean" sensor reports) and as a standalone state estimator.

Bayesian analysis for proportions with an independent background effect

Conventional Bayesian analysis for proportions is not appropriate for proportions with a background effect, for example, for the proportion of correct responses in a sensory difference test using a forced-choice method with a guessing probability. The main difficulty is that it is not reasonable to assume that a proportion with a background effect has a beta prior distribution. A generalized posterior distribution for proportions is derived in this paper. It includes the standard beta posterior distribution for a proportion without background effect as a special case. Bayesian inference and Bayesian sample size determination for proportions based on the generalized posterior distribution are discussed.

A Note on Bayesian Analyses of Capture–Recapture Data with Perfect Recaptures

ABSTRACT A Bayesian analysis of capture–recapture data might assume independent Beta prior distributions for each parameter of interest across competing models. In the absence of genuine a priori knowledge, a default specification would reduce to the special case of uniform priors over the parameter space. In this note we demonstrate that for the set of models proposed by Catchpole et al. (2004), it is possible to derive an exact expression for Bayes factors relating competing models under assumption of unitary recapture probabilities.

Lower System Reliability Bounds from Binary Failure Data Using Bootstrapping

Binary failure data are collected for each of the independent components in a coherent system. Bootstrapping is used to determine a (1 – α)100% lower confidence bound on the system reliability. When a component with perfect test results is encountered, a beta prior distribution is used to avoid an overly optimistic lower bound.

Quantifying experience in sample size determination for drug analysis of seized drugs

Forensic analysis of pills suspected to contain illegal drugs is a time-consuming process; therefore, only a small sample from a seizure can be investigated. Notwithstanding, for drugs like Ecstacy, experience of forensic analysts indicates that a seizure of tablets usually consists either wholly of illicit drugs or no illegal substances at all. Consequently, it should be possible to draw fairly accurate conclusions based on very small samples, if all pills in a sample are indeed analytically identical. The forensic experience is modelled using a beta prior distribution for the proportion of drug-containing tablets in a seizure, and the sample size is determined so that a certain confidence statement can be made about this proportion. The parameters of the beta prior must be set to correspond with the experience, and this paper suggests a method for estimating these parameter values from a database comprising analyst reports representing the experience. The technique is applied to proportions of Ecstacy pills, and the results show that a sample of five pills is enough to state with a high level of confidence that at least half the tablets in a presumed Ecstacy seizure are genuine.

A note on infinite-armed Bernoulli bandit problems with generalized beta prior distributions

A bandit problem with infinitely many Bernoulli arms is considered. The parameters of Bernoulli arms are independent and identically distributed random variables from a generalized beta distributionG3B(a, b, λ) witha, b>0 and 0<λ<2. Under the generalized beta prior distributions, we first derive the asymptotic expected failure rates ofk-failure strategies, and then obtain a lower bound for the expected failure rate over all strategies investigated in Berry et al. (1997). The asymptotic expected failure rates for the other three strategies studied in Berry et al. (1997) are also included. Numerical estimations for a variety of generalized beta prior distributions are presented to illustrate the performances of these strategies.

Efficient Bayesian Sample Size Calculation for Designing a Clinical Trial with Multi‐Cluster Outcome Data

AbstractHealth care utilization and outcome studies call for hierarchical approaches. The objectives were to predict major complications following percutaneous coronary interventions by health providers, and to compare Bayesian and non‐Bayesian sample size calculation methods. The hierarchical data structure consisted of: (1) Strata: PGY4, PGY7, and physician assistant as providers with varied experiences; (2) Clusters: ks providers per stratum; (3) Individuals: ns patients reviewed by each provider. The main outcome event illustrated was mortality modeled by a Bayesian beta‐binomial model. Pilot information and assumptions were utilized to elicit beta prior distributions. Sample size calculations were based on the approximated average length, fixed at 1%, of 95% posterior intervals of the mean event rate parameter. Necessary sample sizes by both non‐Bayesian and Bayesian methods were compared. We demonstrated that the developed Bayesian methods can be efficient and may require fewer subjects to satisfy the same length criterion.

Applications: Posterior Distributions for the Gini Coefficient Using Grouped Data

When available data comprise a number of sampled households in each of a number of income classes, the likelihood function is obtained from a multinomial distribution with the income class population proportions as the unknown parameters. Two methods for going from this likelihood function to a posterior distribution on the Gini coefficient are investigated. In the first method, two alternative assumptions about the underlying income distribution are considered, namely a lognormal distribution and the Singh–Maddala (1976) income distribution. In these cases the likelihood function is reparameterized and the Gini coefficient is a nonlinear function of the income distribution parameters. The Metropolis algorithm is used to find the corresponding posterior distributions of the Gini coefficient from a sample of Bangkok households. The second method does not require an assumption about the nature of the income distribution, but uses (a) triangular prior distributions, and (b) beta prior distributions, on the location of mean income within each income class. By sampling from these distributions, and the Dirichlet posterior distribution of the income class proportions, alternative posterior distributions of the Gini coefficient are calculated.

Some Optimal Strategies for Bandit Problems with Beta Prior Distributions

A bandit problem with infinitely many Bernoulli arms is considered. The parameters of Bernoulli arms are independent and identically distributed random variables from a common distribution with beta(a, b). We investigate the k-failure strategy which is a modification of Robbins's stay-with-a-winner/switch-on-a-loser strategy and three other strategies proposed recently by Berry et al. (1997, Ann. Statist., 25, 2103–2116). We show that the k-failure strategy performs poorly when b is greater than 1, and the best strategy among the k-failure strategies is the 1-failure strategy when b is less than or equal to 1. Utilizing the formulas derived by Berry et al. (1997), we obtain the asymptotic expected failure rates of these three strategies for beta prior distributions. Numerical estimations and simulations for a variety of beta prior distributions are presented to illustrate the performances of these strategies.

BAYESIAN SINGLE-LEVEL BINOMIAL AND EXPONENTIAL RELIABILITY DEMONSTRATION TEST PLANS

Bayesian reliability demonstration test plans are developed for binomial and exponential sampling distributions using mixtures of beta and inverse gamma prior distributions, respectively. Both types of plans consider a single specified target reliability and rectify an important drawback of an existing well-known classical single-level test planning procedure. The plans are developed in such a way that they attain a specified probability of passing the test when the target reliability has been met as well as provide desired confidence that the target reliability has been met given that the test has been passed.

Implementation of Marginal Bayesian Estimation with Four-Parameter Beta Prior Distributions

Item parameter estimation for the threeparameter logistic model (3PLM) is sometimes problematic. The estimation algorithm of the 3PLM maximum likelihood estimation procedure often fails, which results in invalid parameter estimates. A procedure based on the marginal Bayesian estimation method introduced by Bock &amp; Aitkin (1981), Swaminathan &amp; Gifford (1985, 1986), and Mislevy &amp; Bock (1990) is proposed here to improve the item parameter estimates for the 3PLM. Four-parameter beta distributions are used as prior distributions for estimating item parameters. A computer simulation study suggested that implementing the marginal Bayesian estimation algorithm with four-parameter beta prior distributions and then updating the priors with empirical means of the updated intermediate estimates can improve item parameter estimation when accurate prior information about the unknown parameters is not available.

UNCERTAINTY IN BINOMIAL FAILURES AND DEMANDS WITH APPLICATIONS TO RELIABILITY

In estimating a plant-specific binomial probability of failure on demand p in probabilistic risk assessment, the corresponding number of binomial demands n or the observed number of failures x (or both) may be uncertain. We present several methods which account for uncertainties in both x and n when using Bayesian methods to estimate p. A beta prior distribution on p is considered. While the methods formally require the use of numerical integration, approximations are provided to implement them in practice. Several numerical examples are used to illustrate the methods, including a real-data example involving commercial nuclear boiling water reactors.

Empirical Bayes Estimation of the Reliability of Nuclear-Power-Plant Emergency Diesel Generators

For a binomial sampling model using pass/fail data and a beta prior distribution, parametric empirical Bayes (PEB) point and credibility interval estimates are provided for the time-and-diesel average emergency diesel generator (EDG) reliability to load-run on demand at 63 U.S. commercial nuclear power stations. For a geometric sampling model and a beta prior distribution, similar estimates of the time-dependent EDG-average reliability are also obtained for four plants. This is an expository article that introduces seven practical steps, involving both model validation and inference, in applying modern PEB methods. It also illustrates some important advantages of using industry-wide data to make inferences on individual items.

Maximum marginal likelihood estimation for semiparametric item analysis

The item characteristic curve (ICC), defining the relation between ability and the probability of choosing a particular option for a test item, can be estimated by using polynomial regression splines. These provide a more flexible family of functions than is given by the three-parameter logistic family. The estimation of spline ICCs is described by maximizing the marginal likelihood formed by integrating ability over a beta prior distribution. Some simulation results compare this approach with the joint estimation of ability and item parameters.

Bayesian Reliability Analysis of Complex Series/Parallel Systems of Binomial Subsystems and Components

A Bayesian procedure is presented for estimating the reliability (or availability) of a complex system of independent binomial series or parallel subsystems and components. Repeated identical components or subsystems are also permitted. The method uses either test or prior data (perhaps both or neither) at the system, subsystem, and component levels. Beta prior distributions are assumed throughout. The method is motivated and illustrated by the following problem. It is required to estimate the unavailability on demand of the low-pressure coolant injection system in a certain U.S. commercial nuclear-power boiling-water reactor. Three data sources are used to calculate the posterior distribution of the overall system demand unavailability from which the required estimates are obtained. The sensitivity of the results to the three data sources is examined. A FORTRAN computer program for implementing the procedure is available.

Bayesian Reliability Analysis of Complex Series/Parallel Systems of Binomial Subsystems and Components

A Bayesian procedure is presented for estimating the reliability (or availability) of a complex system of independent binomial series or parallel subsystems and components. Repeated identical components or subsystems are also permitted. The method uses either test or prior data (perhaps both or neither) at the system, subsystem, and component levels. Beta prior distributions are assumed throughout. The method is motivated and illustrated by the following problem. It is required to estimate the unavailability on demand of the low-pressure coolant injection system in a certain U.S. commercial nuclear-power boiling-water reactor. Three data sources are used to calculate the posterior distribution of the overall system demand unavailability from which the required estimates are obtained. The sensitivity of the results to the three data sources is examined. A FORTRAN computer program for implementing the procedure is available.

Selecting, under type-II censoring, Weibull populations that are more reliable

It is pointed out that the problem of selecting Weibull populations that are more reliable is complex; the main result is that there is no simple selection rule. Under type-II censoring, the use of a locally optimal selection rule when the shape parameters are known and the use of a modified selection rule when the unknown shape parameters have some prior distributions are proposed. The performance of this modified rule was tested extensively by simulation; this rule was shown to be quite robust for a variety of beta prior distributions. >

Selecting the best binomial population: parametric empirical Bayes approach

Abstract Consider k populations π 1 ,…,π k , where an observation from population π i has a binomial distribution with parameters N and p i (unknown). Let p [ k ] = max 1≤ j ≤ k P j . A population π i with p i = p [ k ] is called a best population. We are interested in selecting the best population. Let p =(p 1 ,…,p k ) and let a denote the index of the selected population. Under the loss function L( p ,a)=p [k] −p a , this statistical selection problem is studied via a parametric empirical Bayes approach. It is assumed that the binomial parameters p i , i =1,…, k , follow some conjugate beta prior distributions with unknown hyperparameters. Under the binomial-beta statistical framework, an empirical Bayes selection rule is proposed. It is shown that the Bayes risk of the proposed empirical Bayes selection rule converges to the corresponding minimum Bayes risk with rates of convergence at least of order O(exp(− cn )) for some positive constant c , where n is the number of accumulated past experience (observations) at hand.