scholarly journals Computation of posterior distribution in Bayesian analysis – application in an intermittently used reliability system

2002 ◽  
Vol 21 (3) ◽  
pp. 78-82
Author(s):  
V. S.S. Yadavalli ◽  
P. J. Mostert ◽  
A. Bekker ◽  
M. Botha
Keyword(s):  
Posterior Distribution ◽  
Jeffreys Prior ◽  
Posterior Density ◽  
Unknown Parameters ◽  
Stationary Rate ◽  
Analysis Application ◽  
Definition Of ◽  
Hpd Intervals ◽  
Highest Posterior Density ◽  
Bayes Procedure

Bayesian estimation is presented for the stationary rate of disappointments, D∞, for two models (with different specifications) of intermittently used systems. The random variables in the system are considered to be independently exponentially distributed. Jeffreys’ prior is assumed for the unknown parameters in the system. Inference about D∞ is being restrained in both models by the complex and non-linear definition of D∞. Monte Carlo simulation is used to derive the posterior distribution of D∞ and subsequently the highest posterior density (HPD) intervals. A numerical example where Bayes estimates and the HPD intervals are determined illustrates these results. This illustration is extended to determine the frequentistical properties of this Bayes procedure, by calculating covering proportions for each of these HPD intervals, assuming fixed values for the parameters.

EXACT BAYESIAN ESTIMATION OF SYSTEM RELIABILITY WITH POTENTIAL MISCLASSIFICATIONS IN SAMPLING

2004 ◽  
Vol 11 (03) ◽  
pp. 223-241
Author(s):  
N. TURKKAN ◽  
T. PHAM-GIA
Keyword(s):  
Bayesian Estimation ◽  
Bayesian Approach ◽  
Posterior Distribution ◽  
System Reliability ◽  
Exact Expression ◽  
System Level ◽  
Posterior Density ◽  
Approximate Methods ◽  
Credible Intervals ◽  
Highest Posterior Density

We provide the exact expression of the reliability of a system under a Bayesian approach, using beta distributions as both native and induced priors at the system level, and allowing uncertainties in sampling, expressed under the form of misclassifications, or noises, that can affect the final posterior distribution. Exact 100(1-α)% highest posterior density credible intervals, for system reliability, are computed, and comparisons are made with results from approximate methods proposed in the literature.

scholarly journals Bayesian Inference on the Shape Parameter and Future Observation of Exponentiated Family of Distributions

2011 ◽  
Vol 2011 ◽  
pp. 1-17
Author(s):  
Sanku Dey ◽  
Sudhansu S. Maiti
Keyword(s):  
Loss Function ◽  
Shape Parameter ◽  
Quadratic Loss ◽  
Posterior Density ◽  
Scale Invariant ◽  
Data Set ◽  
Future Observation ◽  
Hpd Intervals ◽  
Highest Posterior Density ◽  
Family Of Distributions

The Bayes estimators of the shape parameter of exponentiated family of distributions have been derived by considering extension of Jeffreys' noninformative as well as conjugate priors under different scale-invariant loss functions, namely, weighted quadratic loss function, squared-log error loss function and general entropy loss function. The risk functions of these estimators have been studied. We have also considered the highest posterior density (HPD) intervals for the parameter and the equal-tail and HPD prediction intervals for future observation. Finally, we analyze one data set for illustration.

BAYESIAN STUDY OF A TWO-COMPONENT SYSTEM WITH COMMON-CAUSE SHOCK FAILURES

2005 ◽  
Vol 22 (01) ◽  
pp. 105-119 ◽  
Author(s):  
V. S. S. YADAVALLI ◽  
A. BEKKER ◽  
J. PAUW
Keyword(s):  
Steady State ◽  
Posterior Density ◽  
Two Component System ◽  
Unknown Parameters ◽  
Component System ◽  
Common Cause ◽  
Two Component ◽  
Highest Posterior Density Intervals ◽  
In Series ◽  
Highest Posterior Density

The steady-state availability of a two-component system in series and parallel subject to individual failures (I-failures) and common-cause shock (CCS) failures is studied from a Bayesian viewpoint with different types of priors assumed for the unknown parameters in the system. Monte Carlo simulation is used to derive the posterior distribution for the steady-state availability and subsequently the highest posterior density intervals. A numerical example illustrates the results.

Revisit to progressively Type-II censored competing risks data from Lomax distributions

2021 ◽  
pp. 1748006X2098397
Author(s):  
Rui Hua ◽  
Wenhao Gui
Keyword(s):  
Confidence Intervals ◽  
Competing Risks ◽  
Maximum Likelihood Estimates ◽  
Type Ii ◽  
Posterior Density ◽  
Unknown Parameters ◽  
Linex Loss ◽  
Credible Intervals ◽  
Competing Risks Data ◽  
Highest Posterior Density

In survival analysis, more than one factor typically contributes to individual failure. In addition, censoring is inevitable in lifespan tests or reliability studies due to external causes or experimental purposes. In this article, the competing risks model is considered and investigated under progressively Type-II censoring where data is from Lomax distributions. Assumptions are further made that these competitive factors are independently distributed, and the latent lifetimes of these factors follow Lomax distributions where both scale parameters and shape parameters are different. For all unknown parameters, maximum likelihood estimates have been attained by Newton-Raphson (NR) method as well as expectation maximization (EM) method, and then the approximate confidence intervals are acquired, in addition to bootstrap confidence intervals. Furthermore, under square error and LINEX loss functions, Bayes estimates and corresponding highest posterior density credible intervals are successively constructed. Finally, simulation experiments are implemented to access performance of several proposed methods in this article, and laboratory dataset is presented and analyzed for illustrative purposes.

scholarly journals Classical and Bayesian Inference of an Exponentiated Half-Logistic Distribution under Adaptive Type II Progressive Censoring

Entropy ◽  
2021 ◽  
Vol 23 (12) ◽  
pp. 1558
Author(s):  
Ziyu Xiong ◽  
Wenhao Gui
Keyword(s):  
Confidence Intervals ◽  
Bootstrap Method ◽  
Information Matrix ◽  
Logistic Distribution ◽  
Type Ii ◽  
Progressive Censoring ◽  
Posterior Density ◽  
Unknown Parameters ◽  
Data Set ◽  
Highest Posterior Density

The point and interval estimations for the unknown parameters of an exponentiated half-logistic distribution based on adaptive type II progressive censoring are obtained in this article. At the beginning, the maximum likelihood estimators are derived. Afterward, the observed and expected Fisher’s information matrix are obtained to construct the asymptotic confidence intervals. Meanwhile, the percentile bootstrap method and the bootstrap-t method are put forward for the establishment of confidence intervals. With respect to Bayesian estimation, the Lindley method is used under three different loss functions. The importance sampling method is also applied to calculate Bayesian estimates and construct corresponding highest posterior density (HPD) credible intervals. Finally, numerous simulation studies are conducted on the basis of Markov Chain Monte Carlo (MCMC) samples to contrast the performance of the estimations, and an authentic data set is analyzed for exemplifying intention.

scholarly journals Sample Size Requirements for Calibrated Approximate Credible Intervals for Proportions in Clinical Trials

2021 ◽  
Vol 18 (2) ◽  
pp. 595
Author(s):  
Fulvio De Santis ◽  
Stefania Gubbiotti
Keyword(s):  
Clinical Trials ◽  
Posterior Probability ◽  
Interval Estimation ◽  
Small Sample ◽  
Sample Sizes ◽  
Posterior Density ◽  
Unknown Parameters ◽  
Credible Intervals ◽  
Small Sample Sizes ◽  
Highest Posterior Density

In Bayesian analysis of clinical trials data, credible intervals are widely used for inference on unknown parameters of interest, such as treatment effects or differences in treatments effects. Highest Posterior Density (HPD) sets are often used because they guarantee the shortest length. In most of standard problems, closed-form expressions for exact HPD intervals do not exist, but they are available for intervals based on the normal approximation of the posterior distribution. For small sample sizes, approximate intervals may be not calibrated in terms of posterior probability, but for increasing sample sizes their posterior probability tends to the correct credible level and they become closer and closer to exact sets. The article proposes a predictive analysis to select appropriate sample sizes needed to have approximate intervals calibrated at a pre-specified level. Examples are given for interval estimation of proportions and log-odds.

scholarly journals Adaptive Gaussian Process Approximation for Bayesian Inference with Expensive Likelihood Functions

2018 ◽  
Vol 30 (11) ◽  
pp. 3072-3094 ◽  
Author(s):  
Hongqiao Wang ◽  
Jinglai Li
Keyword(s):  
Bayesian Inference ◽  
Gaussian Process ◽  
Adaptive Algorithm ◽  
Posterior Density ◽  
Unknown Parameters ◽  
Likelihood Functions ◽  
Inference Problems ◽  
Active Learning Method ◽  
Computationally Intensive ◽  
Process Approximation

We consider Bayesian inference problems with computationally intensive likelihood functions. We propose a Gaussian process (GP)–based method to approximate the joint distribution of the unknown parameters and the data, built on recent work (Kandasamy, Schneider, & Póczos, 2015 ). In particular, we write the joint density approximately as a product of an approximate posterior density and an exponentiated GP surrogate. We then provide an adaptive algorithm to construct such an approximation, where an active learning method is used to choose the design points. With numerical examples, we illustrate that the proposed method has competitive performance against existing approaches for Bayesian computation.

scholarly journals The Confidence Density for Correlation

Sankhya A ◽  
2021 ◽  
Author(s):  
Gunnar Taraldsen
Keyword(s):  
Explicit Formula ◽  
Posterior Distribution ◽  
Hypergeometric Functions ◽  
Real Data ◽  
Numerical Calculations ◽  
Posterior Density ◽  
Fiducial Inference ◽  
Objective Prior ◽  
Confidence Distribution ◽  
New Formula

AbstractInference for correlation is central in statistics. From a Bayesian viewpoint, the final most complete outcome of inference for the correlation is the posterior distribution. An explicit formula for the posterior density for the correlation for the binormal is derived. This posterior is an optimal confidence distribution and corresponds to a standard objective prior. It coincides with the fiducial introduced by R.A. Fisher in 1930 in his first paper on fiducial inference. C.R. Rao derived an explicit elegant formula for this fiducial density, but the new formula using hypergeometric functions is better suited for numerical calculations. Several examples on real data are presented for illustration. A brief review of the connections between confidence distributions and Bayesian and fiducial inference is given in an Appendix.

scholarly journals Efeito materno e paterno sobre as taxas de fertilização e eclosão em curimba (Prochilodus lineatus)

2012 ◽  
Vol 64 (6) ◽  
pp. 1584-1590 ◽  
Author(s):  
I.B. Allaman ◽  
R.T.F. Freitas ◽  
A.T.M. Viveiros ◽  
A.F. Nascimento ◽  
G.R. Oliveira ◽  
...  
Keyword(s):  
Posterior Density ◽  
Prochilodus Lineatus ◽  
Highest Posterior Density

Avaliou-se o quanto fêmeas e machos contribuem para a variação total das taxas de fertilização e de eclosão em curimba (Prochilodus lineatus). Utilizou-se sêmen criopreservado proveniente de cinco machos para fertilizar ovócitos de seis fêmeas em um esquema fatorial cruzado 5x6, totalizando 30 famílias. Além das características reprodutivas dos machos e fêmeas, foram avaliadas as taxas de fertilização e eclosão para cômputo dos efeitos materno e paterno. Os componentes da variância foram estimados por meio da máxima verossimilhança restrita, sendo construídos intervalos Highest Posterior Density (HPD) para cada componente. Verificou-se que as fêmeas contribuíram muito mais para a variação total em relação aos machos para as taxas de fertilização e eclosão. Para a taxa de fertilização, as fêmeas contribuíram com 26,3% da variação total e os machos com 8,9%. Em relação à taxa de eclosão, as fêmeas contribuíram com 11,9% e os machos com 1,6%. Concluiu-se que houve efeito materno sobre as taxas de fertilização e eclosão e que o efeito paterno avaliado individualmente foi pouco expressivo ou até mesmo insignificante.

scholarly journals Reliable confidence intervals for RelTime estimates of evolutionary divergence times

2019 ◽  
Author(s):  
Qiqing Tao ◽  
Koichiro Tamura ◽  
Beatriz Mello ◽  
Sudhir Kumar
Keyword(s):  
Confidence Intervals ◽  
Divergence Time ◽  
Simulated Data ◽  
Molecular Dating ◽  
Divergence Times ◽  
Rate Variation ◽  
Evolutionary Divergence ◽  
Posterior Density ◽  
Divergence Time Estimates ◽  
Highest Posterior Density

AbstractConfidence intervals (CIs) depict the statistical uncertainty surrounding evolutionary divergence time estimates. They capture variance contributed by the finite number of sequences and sites used in the alignment, deviations of evolutionary rates from a strict molecular clock in a phylogeny, and uncertainty associated with clock calibrations. Reliable tests of biological hypotheses demand reliable CIs. However, current non-Bayesian methods may produce unreliable CIs because they do not incorporate rate variation among lineages and interactions among clock calibrations properly. Here, we present a new analytical method to calculate CIs of divergence times estimated using the RelTime method, along with an approach to utilize multiple calibration uncertainty densities in these analyses. Empirical data analyses showed that the new methods produce CIs that overlap with Bayesian highest posterior density (HPD) intervals. In the analysis of computer-simulated data, we found that RelTime CIs show excellent average coverage probabilities, i.e., the true time is contained within the CIs with a 95% probability. These developments will encourage broader use of computationally-efficient RelTime approach in molecular dating analyses and biological hypothesis testing.

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