Conjugate Prior Research Articles

On Bayesian Estimation of Loss and Risk Functions

Loss functions and Risk functions play very important role in Bayesian estimation. This paper aims at the Bayesian estimation for the loss and risk functions of the unknown parameter of the H(r, theta), (theta being the unknown parameter) distribution The estimation has been performed under Rukhin’s loss function. The importance of this distribution is that it contains some important distributions such as the Half Normal distribution, Rayleigh distribution and Maxwell’s distribution as particular cases. The inverse Gamma distribution has assumed as the prior distribution for the unknown parameter theta. This prior distribution is a Natural Conjugate prior distribution for the unknown parameter because the posterior probability density function of the unknown parameter is also inverse gamma distribution The Rukhin’s loss function involves another loss function denoted by w(theta, delta) he form of w(theta, delta) is important as it changes the estimate. In this paper, three forms of w(theta, delta) have been taken and corresponding estimates have been derived. The three, forms are, the Squared Error Loss Function (SELF) and two different forms of Weighted Squared Error Loss Function (WSELF) namely, the Minimum Expected Loss (MELO) Function and the Exponentially Weighted Minimum Expected Loss (EWMELO) Function have been considered. A criterion of performance of various form of w(theta, delta) has ben defined. It has been proved that among three forms of w(theta, delta), considered here, the form corresponding to EWMELO is most dominant.

E‐Bayesian estimation of reliability characteristics of two‐parameter bathtub‐shaped lifetime distribution with application

AbstractThis article presents the expected Bayesian (E‐Bayesian) estimation of the scale parameter, reliability and failure rate functions of two‐parameter bathtub‐shaped lifetime distribution under type‐II censoring data with. Squared error loss function and gamma distribution as a conjugate prior distribution for the unknown parameter are used to obtain the E‐Bayesian estimators. Also, three different prior distributions for the hyperparameters for the E‐Bayesian estimators are considered. Some properties of the E‐Bayesian estimators are studied. Using minimum mean square error criteria, a simulation study is conducted to compare the performance of the E‐Bayesian estimators and the corresponding Bayes and maximum likelihood estimators. A real data set is analysed to show the applicability of the different proposed estimators. The numerical results show that the E‐Bayesian estimators perform better than the classical and Bayesian estimators.

A mathematical model of local and global attention in natural scene viewing.

Understanding the decision process underlying gaze control is an important question in cognitive neuroscience with applications in diverse fields ranging from psychology to computer vision. The decision for choosing an upcoming saccade target can be framed as a selection process between two states: Should the observer further inspect the information near the current gaze position (local attention) or continue with exploration of other patches of the given scene (global attention)? Here we propose and investigate a mathematical model motivated by switching between these two attentional states during scene viewing. The model is derived from a minimal set of assumptions that generates realistic eye movement behavior. We implemented a Bayesian approach for model parameter inference based on the model’s likelihood function. In order to simplify the inference, we applied data augmentation methods that allowed the use of conjugate priors and the construction of an efficient Gibbs sampler. This approach turned out to be numerically efficient and permitted fitting interindividual differences in saccade statistics. Thus, the main contribution of our modeling approach is two–fold; first, we propose a new model for saccade generation in scene viewing. Second, we demonstrate the use of novel methods from Bayesian inference in the field of scan path modeling.

Variational Bayes Inference Algorithm for the Saturated Diagnostic Classification Model.

Saturated diagnostic classification models (DCM) can flexibly accommodate various relationships among attributes to diagnose individual attribute mastery, and include various important DCMs as sub-models. However, the existing formulations of the saturated DCM are not better suited for deriving conditionally conjugate priors of model parameters. Because their derivation is the key in developing a variational Bayes (VB) inference algorithm, in the present study, we proposed a novel mixture formulation of saturated DCM. Based on it, we developed a VB inference algorithm of the saturated DCM that enables us to perform scalable and computationally efficient Bayesian estimation. The simulation study indicated that the proposed algorithm could recover the parameters in various conditions. It has also been demonstrated that the proposed approach is particularly suited to the case when new data become sequentially available over time, such as in computerized diagnostic testing. In addition, a real educational dataset was comparatively analyzed with the proposed VB and Markov chain Monte Carlo (MCMC) algorithms. The result demonstrated that very similar estimates were obtained between the two methods and that the proposed VB inference was much faster than MCMC. The proposed method can be a practical solution to the problem of computational load.

A Bayesian analysis of the matching problem

The matching problem is known since the beginning of the eighteenth century and Bayesian solutions have been proposed. In this paper, we present a Bayesian analysis of an experiment that also leads to the matching problem. Since in this paper we consider the order in which assignments are made and not only the number of matches, our approach is different from the literature on this problem. Our approach also considers that it is possible to have different abilities for the different objects that will be guessed. Hence, we have a parameter that measures the matching ability for each specific object or assignment. As a consequence, we need to have some replicates of the experiment and a prior distribution for each of the parameters. Conjugate prior distributions for the parameters are discussed. The frequentist solution has a particular Bayesian interpretation under a non-informative prior distribution. A real data set is analyzed using the proposed methodology.

Estimating Common Scale Parameter of Two Logistic Populations: A Bayesian Study

Estimation under equality restrictions is an age old problem and has been considered by several researchers in the past due to practical applications and theoretical challenges involved in it. Particularly, the problem has been extensively studied from classical as well as decision theoretic point of view when the underlying distribution is normal. In this paper, we consider the problem when the underlying distribution is non-normal, say, logistic. Specifically, estimation of the common scale parameter of two logistic populations has been considered when the location parameters are unknown. It is observed that closed forms of the maximum likelihood estimators (MLEs) for the associated parameters do not exist. Using certain numerical techniques the MLEs have been derived. The asymptotic confidence intervals have been derived numerically too, as these also depend on the MLEs. Approximate Bayes estimators are proposed using non-informative as well as conjugate priors with respect to the squared error (SE) and the LINEX loss functions. A simulation study has been conducted to evaluate the proposed estimators and compare their performances through mean squared error (MSE) and bias. Finally, two real life examples have been considered in order to show the potential applications of the proposed model and illustrate the method of estimation.

VoI-informed decision-making for SHM system arrangement

Structural health monitoring systems have been widely implemented to provide real-time continuous data support and to ensure structural safety in the context of structural integrity management. However, the quantification of the potential benefits of structural health monitoring systems has not yet attracted widespread attention. At the same time, there is an urgent need to develop strategies, such as optimizing the monitoring period, monitoring variables, and other factors, to maximize the potential benefits of structural health monitoring systems. Considering the continuity of structural health monitoring information, a framework is developed in this article to support decision-making for structural health monitoring systems arrangement in the context of structural integrity management, which integrates the concepts of value of information and risk-based inspection planning based on an approach which utilizes a conjugate prior probability distribution for updating of the probabilistic models of structural performances based on structural health monitoring information. An example considering fatigue degradation of steel structures is investigated to illustrate the application of the proposed framework. The considered example shows that the choice of monitoring variables, the monitoring period, and the monitoring quality may be consistently optimized by the application of the proposed framework and approach. Finally, discussions and conclusions are provided to clarify the potential benefits of the proposed framework with a special view to practical applications of structural health monitoring systems.

Efficient integration of generative topic models into discriminative classifiers using robust probabilistic kernels

We propose an alternative to the generative classifier that usually models both the class conditionals and class priors separately, and then uses the Bayes theorem to compute the posterior distribution of classes given the training set as a decision boundary. Because SVM (support vector machine) is not a probabilistic framework, it is really difficult to implement a direct posterior distribution-based discriminative classifier. As SVM lacks in full Bayesian analysis, we propose a hybrid (generative–discriminative) technique where the generative topic features from a Bayesian learning are fed to the SVM. The standard latent Dirichlet allocation topic model with its Dirichlet (Dir) prior could be defined as Dir–Dir topic model to characterize the Dirichlet placed on the document and corpus parameters. With very flexible conjugate priors to the multinomials such as generalized Dirichlet (GD) and Beta-Liouville (BL) in our proposed approach, we define two new topic models: the BL–GD and GD–BL. We take advantage of the geometric interpretation of our generative topic (latent) models that associate a K-dimensional manifold (K is the size of the topics) embedded into a V-dimensional feature space (word simplex) where V is the vocabulary size. Under this structure, the low-dimensional topic simplex (the subspace) defines a document as a single point on its manifold and associates each document with a single probability. The SVM, with its kernel trick, performs on these document probabilities in classification where it utilizes the maximum margin learning approach as a decision boundary. The key note is that points or documents that are close to each other on the manifold must belong to the same class. Experimental results with text documents and images show the merits of the proposed framework.

Gibbs-Slice Sampling Algorithm for Estimating the Four-Parameter Logistic Model

The four-parameter logistic (4PL) model has recently attracted much interest in educational testing and psychological measurement. This paper develops a new Gibbs-slice sampling algorithm for estimating the 4PL model parameters in a fully Bayesian framework. Here, the Gibbs algorithm is employed to improve the sampling efficiency by using the conjugate prior distributions in updating asymptote parameters. A slice sampling algorithm is used to update the 2PL model parameters, which overcomes the dependence of the Metropolis–Hastings algorithm on the proposal distribution (tuning parameters). In fact, the Gibbs-slice sampling algorithm not only improves the accuracy of parameter estimation, but also enhances sampling efficiency. Simulation studies are conducted to show the good performance of the proposed Gibbs-slice sampling algorithm and to investigate the impact of different choices of prior distribution on the accuracy of parameter estimation. Based on Markov chain Monte Carlo samples from the posterior distributions, the deviance information criterion and the logarithm of the pseudomarginal likelihood are considered to assess the model fittings. Moreover, a detailed analysis of PISA data is carried out to illustrate the proposed methodology.

Latent Dirichlet Analysis of Categorical Survey Responses

Beliefs are important determinants of an individual’s choices and economic outcomes, so understanding how they comove and differ across individuals is of considerable interest. Researchers often rely on surveys that report individual beliefs as qualitative data. We propose using a Bayesian hierarchical latent class model to analyze the comovements and observed heterogeneity in categorical survey responses. We show that the statistical model corresponds to an economic structural model of information acquisition, which guides interpretation and estimation of the model parameters. An algorithm based on stochastic optimization is proposed to estimate a model for repeated surveys when responses follow a dynamic structure and conjugate priors are not appropriate. Guidance on selecting the number of belief types is also provided. Two examples are considered. The first shows that there is information in the Michigan survey responses beyond the consumer sentiment index that is officially published. The second shows that belief types constructed from survey responses can be used in a subsequent analysis to estimate heterogeneous returns to education.

Conjugate Priors and Posterior Inference for the Matrix Langevin Distribution on the Stiefel Manifold

Directional data emerges in a wide array of applications, ranging from atmospheric sciences to medical imaging. Modeling such data, however, poses unique challenges by virtue of their being constrained to non-Euclidean spaces like manifolds. Here, we present a unified Bayesian framework for inference on the Stiefel manifold using the Matrix Langevin distribution. Specifically, we propose a novel family of conjugate priors and establish a number of theoretical properties relevant to statistical inference. Conjugacy enables translation of these properties to their corresponding posteriors, which we exploit to develop the posterior inference scheme. For the implementation of the posterior computation, including the posterior sampling, we adopt a novel computational procedure for evaluating the hypergeometric function of matrix arguments that appears as normalization constants in the relevant densities.

Estimation of the Parameters for the Exponentiated Weibull Distribution Based on Progressive Type-II Censoring Scheme

The main objective of the present study is to find the estimation of the two Exponentiated Weibull distribution parameters, based on progressive Type II censored samples. The maximum likelihood and Bayes estimators for the two shape parameters and the scale parameter of the exponentiated Weibull lifetime model were derived. Bayes estimators was obtained by using both the symmetric and asymmetric loss functions via squared error loss and linex loss functions This was done with respect to the conjugate priors for two shape parameters. We used an approximation based on the Lindley (Trabajos de Estadistca) method for obtaining Bayes estimates under these loss functions. The different proposed estimators have been compared through an extensive simulation studies. Bayes ratings also turned out to be better than MLE. Whatever the sample sizes are, we get the same results.

Statistical inference of Type-I progressively censored step-stress accelerated life test with dependent competing risks

This paper considers a step-stress accelerated dependent competing risks model under progressively Type-I censoring schemes. The dependence structure between competing risks is modeled by a general bivariate function, the cumulative exposure model is assumed and the accelerated model is described by the power rule model. The point and interval estimation of the model parameters and the reliability under normal usage level at mission time are obtained by using the maximum likelihood method and the asymptotic normal theory. We also consider the Bayesian estimators and the highest posterior density credible intervals based on conjugate priors, E-Bayesian, hierarchical Bayesian and empirical Bayesian methods. To illustrate the proposed methodology, the Marshall-Olkin bivariate exponential distribution is used to model the dependence structure between competing risks. A Monte Carlo simulation study and a real data analysis are presented to study the performance of different estimation methods.

Inferences for Weibull parameters under progressively first-failure censored data with binomial random removals

In this paper, the Bayesian and non-Bayesian estimation of a two-parameter Weibull lifetime model in presence of progressive first-failure censored data with binomial random removals are considered. Based on the s-normal approximation to the asymptotic distribution of maximum likelihood estimators, two-sided approximate confidence intervals for the unknown parameters are constructed. Using gamma conjugate priors, several Bayes estimates and associated credible intervals are obtained relative to the squared error loss function. Proposed estimators cannot be expressed in closed forms and can be evaluated numerically by some suitable iterative procedure. A Bayesian approach is developed using Markov chain Monte Carlo techniques to generate samples from the posterior distributions and in turn computing the Bayes estimates and associated credible intervals. To analyze the performance of the proposed estimators, a Monte Carlo simulation study is conducted. Finally, a real data set is discussed for illustration purposes.

Bayesian Inference on the Generalized Gamma Distribution using Conjugate Priors

This paper focuses on the three-parameter generalized gamma distribution and uses Bayesian techniques to estimate its parameters. Many authors con-sidered estimating the parameters of the generalized gamma distribution in a Bayesian framework using Jeffrey’s priors. Others used different loss functions and the least squares approach. This study uses Bayesian techniques to estimate the three-parameter generalized gamma distribution by using conjugate priors. The random Metropolis algorithm is used to simulate the Bayesian estimates of the three parameters. Then these estimates are compared to the maximum like-lihood estimates using the mean error through simulation. It has been shown in this paper that the obtained estimates using this approach is more accurate than the traditional methods of estimation such as the Maximum likelihood method. The same approach is then used to estimate the parameters of mixtures of the generalized gamma parameters using conjugate priors.

A Review on Inverse Maxwell Distribution with Its Statistical Properties and Applications

In this article, we review the inverse Maxwell distribution and establish its statistical properties including moments, descriptive measures and stochastic orderings. We derive complete sufficient statistic, minimum variance unbiased estimator and maximum likelihood estimator for the parameter. We obtain Bayes estimators of the parameter under non-informative (Jeffrey’s) as well as conjugate priors. We also provide confidence and highest posterior density intervals for the parameter. Two real data sets, GAGurine and Guinea pig data, have been considered for numerical illustrations. Finally, we discuss some areas of applications and further extensions of inverse Maxwell distribution for the future research work.

Bayesian structural equation modeling in multiple omics data with application to circadian genes

It is well known that the integration among different data-sources is reliable because of its potential of unveiling new functionalities of the genomic expressions, which might be dormant in a single-source analysis. Moreover, different studies have justified the more powerful analyses of multi-platform data. Toward this, in this study, we consider the circadian genes' omics profile, such as copy number changes and RNA-sequence data along with their survival response. We develop a Bayesian structural equation modeling coupled with linear regressions and log normal accelerated failure-time regression to integrate the information between these two platforms to predict the survival of the subjects. We place conjugate priors on the regression parameters and derive the Gibbs sampler using the conditional distributions of them. Our extensive simulation study shows that the integrative model provides a better fit to the data than its closest competitor. The analyses of glioblastoma cancer data and the breast cancer data from TCGA, the largest genomics and transcriptomics database, support our findings. The developed method is wrapped in R package available at https://github.com/MAITYA02/semmcmc. Supplementary data are available at Bioinformatics online.

Bayesian adaptive design for concurrent trials involving biologically related diseases.

We develop a Bayesian design method for a clinical program where an investigational product is to be studied concurrently in a set of clinical trials involving related diseases with the goal of demonstrating superiority to a control in each. The approach borrows information on treatment effectiveness using correlated mixture priors using an analysis procedure that is closely related Bayesian model averaging. Mixture priors are constructed by eliciting conjugate priors based on pessimistic and enthusiastic predictions for the data to be observed for each disease and then by eliciting mixture weights for all possible configurations of the pessimistic and enthusiastic priors across the diseases to be studied. The proposed approach provides a robust framework for information borrowing in settings where the diseases may have endpoints based on different data types. We show via simulation that operating characteristics based on the proposed design framework are favorable compared to those based on information borrowing designs using the Bayesian hierarchical model which is poorly suited for information borrowing when there are different data types underpinning the endpoints across which information is to be borrowed.

A generalized framework for classical test theory

This paper develops a generalized framework which allows for the use of parametric classical test theory inference with non-normal models. Using the theory of natural exponential families and Bayesian theory of their conjugate priors, theoretical properties of test scores under the framework are derived, including a formula for parallel-test reliability in terms of the test length and a parameter of the underlying population distribution of abilities. This framework is shown to satisfy the general properties of classical test theory several common classical test theory results are shown to reduce to parallel-test reliability in this framework. An empirical Bayes method for estimating reliability, both with point estimates and with intervals, is described using maximum likelihood. This method is applied to an example set of data and compared to classical test theory estimators of reliability, and a simulation study is performed to show the coverage of the interval estimates of reliability derived from the framework.

An improved Bayesian Modified-EWMA location chart and its applications in mechanical and sport industry.

Control charts are popular tools in the statistical process control toolkit and the exponentially weighted moving average (EWMA) chart is one of its essential component for efficient process monitoring. In the present study, a new Bayesian Modified-EWMA chart is proposed for the monitoring of the location parameter in a process. Four various loss functions and a conjugate prior distribution are used in this study. The average run length is used as a performance evaluation tool for the proposed chart and its counterparts. The results advocate that the proposed chart performs very well for the monitoring of small to moderate shifts in the process and beats the existing counterparts. The significance of the proposed scheme has proved through two real-life examples: (1) For the monitoring of the reaming process which is used in the mechanical industry. (2) For the monitoring of golf ball performance in the sports industry.