Bayesian sensitivity analysis in elliptical linear regression models

The objective of this study is to examine the properties of Bayes estimators of the shape parameter of the Power Function Distribution (PFD-I), by using two different prior distributions for the parameter θ and different loss functions that were compared with the maximum likelihood estimators. In many practical applications, we may have two different prior information about the prior distribution for the shape parameter of the Power Function Distribution, which influences the parameter estimation. So, we used two different kinds of conjugate priors of shape parameter θ of the Power Function Distribution (PFD-I) to estimate it. The conjugate prior function of the shape parameter θ was considered as a combination of two different prior distributions such as gamma distribution with Erlang distribution and Erlang distribution with exponential distribution and Erlang distribution with non-informative distribution and exponential distribution with the non-informative distribution. We derived Bayes estimators for shape parameter θ of the Power Function Distribution (PFD-I) according to different loss functions such as the squared error loss function (SELF), the weighted error loss function (WSELF) and modified linear exponential (MLINEX) loss function (MLF), with two different double priors. In addition to the classical estimation (maximum likelihood estimation). We used simulation to get the results of this study, for different cases of the shape parameter of the Power Function Distribution used to generate data for different samples sizes.

This paper proposed a Bayesian method-based structural equation model (SEM) of miners’ work injury for an underground coal mine in India. The environmental and behavioural variables for work injury were identified and causal relationships were developed. For Bayesian modelling, prior distributions of SEM parameters are necessary to develop the model. In this paper, two approaches were adopted to obtain prior distribution for factor loading parameters and structural parameters of SEM. In the first approach, the prior distributions were considered as a fixed distribution function with specific parameter values, whereas, in the second approach, prior distributions of the parameters were generated from experts’ opinions. The posterior distributions of these parameters were obtained by applying Bayesian rule. The Markov Chain Monte Carlo sampling in the form Gibbs sampling was applied for sampling from the posterior distribution. The results revealed that all coefficients of structural and measurement model parameters are statistically significant in experts’ opinion-based priors, whereas, two coefficients are not statistically significant when fixed prior-based distributions are applied. The error statistics reveals that Bayesian structural model provides reasonably good fit of work injury with high coefficient of determination (0.91) and less mean squared error as compared to traditional SEM.

An autoregressive moving average (ARMA) is a time series model that is applied in everyday life for pattern recognition and forecasting. The ARMA model contains a noise which is assumed to have a specific distribution. The noise is often considered to have a Gaussian distribution. However in applications, the noise is sometimes found that does not have a Gaussian distribution. The first objective is to develop the ARMA model in which noise has a Laplacian distribution. The second objective is to estimate the parameters of the ARMA model. The ARMA model parameters include ARMA model orders, ARMA model coefficients, and noise variance. The parameter estimation of the ARMA model is carried out in the Bayesian framework. In the Bayesian framework, the ARMA model parameters are treated as a variable that has a prior distribution. The prior distribution for the ARMA model parameters is combined with the likelihood function for the data to get the posterior distribution for the parameter. The posterior distribution for parameters has a complex form so that the Bayes estimator cannot be determined analytically. The reversible jump Markov chain Monte Carlo (MCMC) algorithm was adopted to determine the Bayes estimator. The first result, the ARMA model can be developed by assuming Laplacian distribution noise. The second result, the performance of the algorithm was tested using simulation studies. The simulation shows that the reversible jump MCMC algorithm can estimate the parameters of the ARMA model correctly.

In practice, it is desired to have estimates that are invariant under reparameterization. The invariance property of the estimators helps to formulate a unified solution to the underlying estimation problem. In robust Bayesian analysis, a frequent criticism is that the optimal estimators are not invariant under smooth reparameterizations. This paper considers the problem of posterior regret gamma-minimax (PRGM) estimation of the natural parameter of the exponential family of distributions under intrinsic loss functions with Kullback-Leibler distance. We show that under the class of Jeffrey’s Conjugate Prior (JCP) distributions, PRGM estimators are invariant to smooth one-to-one reparameterizations. We apply our results to several distributions and different classes of JCP, as well as the usual conjugate prior distributions. We observe that, in many cases, invariant PRGM estimators in the class of JCP distributions can be obtained by some modifications of PRGM estimators in the usual class of conjugate priors. Moreover, when the class of priors are convex or dependant on a hyper-parameter belonging to a connected set, we show that the PRGM estimator under the intrinsic loss function could be Bayes with respect to a prior distribution in the original prior class. Theoretical results are supplemented with several examples and illustrations.

This article discusses the Bayesian approach for count data using non-homogeneous Poisson processes, considering different prior distributions for the model parameters. A Bayesian approach using Markov Chain Monte Carlo (MCMC) simulation methods for this model was first introduced by [1], taking into account software reliability data and considering non-informative prior distributions for the parameters of the model. With the non-informative prior distributions presented by these authors, computational difficulties may occur when using MCMC methods. This article considers different prior distributions for the parameters of the proposed model, and studies the effect of such prior distributions on the convergence and accuracy of the results. In order to illustrate the proposed methodology, two examples are considered: the first one has simulated data, and the second has a set of data for pollution issues at a region in Mexico City.

Thermodynamic modeling and uncertainty quantification of CO2-loaded aqueous MEA solutions

Practical use of Bayesian estimation procedures is often associated with difficulties related to elicitation of prior information, and its formalization into the respective prior distribution. The two-parameter Weibull distribution is a particularly difficult case, because it requires a two-dimensional joint prior distribution of the Weibull parameters. The novelty of the procedure suggested here is that the prior information can be presented in the form of the interval assessment of the reliability function (as opposed to that on the Weibull parameters), which is generally easier to obtain. Based on this prior information, the procedure allows constructing the continuous joint prior distribution of Weibull parameters as well as the posterior estimates of the mean and standard deviation of the estimated reliability function (or the CDF) at any given value of the exposure variable. A numeric example is discussed as an illustration. We additionally elaborate on a new parametric form of the prior distribution for the scale parameter of the exponential distribution. This distribution is not a Gamma (as might intuitively be expected); its mode is available in a closed form, and the mean is obtainable through a series approximation.

PurposeThe purpose of this paper is to present an economic–statistical design (ESD) for the Bayesian control chart based on predictive distribution with two types of informative and noninformative prior distributions.Design/methodology/approachThe design used in this study is based on determining the control chart of the predictive distribution and then its ESD. The new proposed cost model is presented by considering the conjugate and Jeffrey's prior distribution in calculating the expected total cycle time and expected cost per cycle, and finally, the optimal design parameters and related costs are compared with the fixed ratio sampling (FRS) mode.FindingsNumerical results show decreases in costs in this Bayesian approach with both Jeffrey's and conjugate prior distribution compared to the FRS mode. This result shows that the Bayesian approach which is based on predictive density works better than the classical approach. Also, for the Bayesian approach, however, there is no significant difference between the results of using Jeffrey's and conjugate prior distributions. Using sensitivity analysis, the effect of cost parameters and shock model parameters and deviation from the mean on the optimal values of design parameters and related costs have been investigated and discussed.Practical implicationsThis research adds to the body of knowledge related to quality control of process monitoring systems. This paper may be of particular interest to quality system practitioners for whom the effect of the prior distribution of parameters on the quality characteristic distribution is important.Originality/valueeconomic statistical design (ESD) of Bayesian control charts based on predictive distribution is presented for the first time.

Bayesian inference for normal regression models, including sensitivity analysis, model comparison and error in variables under non-informative and conjugate prior for the model parameters has received considerable attention in the last decades. From a distributional point of view the results can be extended in several directions. One is by considering a wider class of prior distributions for the parameters of the model. Another, is by considering alternative distributions for the error terms. Usually, the results with non-conjugate priors rely heavily on MCMC methods. On the other hand, many extensions has been obtained by considering the so called dependent elliptical model, which is often used in linear regression analysis to accommodate the kurtosis of the error terms and to accommodate outliers. Bayesian inference with multivariate elliptical models was initially presented in Chu (1973). Posteriorly, Meinhold and Singpurwalla (1989) consider a robustification of Kalman filters by using the multivariate Student-t distribution. This distribution was also used by Zellner (1976), who considered a Bayesian treatment of linear regression models under non-informative prior distributions.

In the decades since Markov chain Monte Carlo methods were first introduced, they have revolutionised Bayesian approaches to statistical inference. Each new advance in MCMC methodology produces near immediate benefits for Bayesian practitioners, expanding the range of problems they can feasibly solve. In this paper, we explore ways in which Bayesian approaches can return something of the debt owed to MCMC, by using explicitly Bayesian concepts to aid in the design of MCMC samplers. The art of efficient MCMC sampling lies in designing a Markov process that (a) has the required limiting distribution, (b) has good convergence and mixing properties and (c) can be implemented in a computationally efficient manner. In this paper, we explore the idea that the selection of an appropriate process, and in particular the tuning of the parameters of the process to achieve the above goals, can be regarded as a problem of estimation. As such, it is amenable to a conventional Bayesian approach, in which a prior distribution for optimal parameters of the sampler is specified, data relevant to sampler performance is obtained and a posterior distribution for optimal parameters is formed. Sampling from this posterior distribution can then be incorporated into the MCMC sampler to produce an adaptive method. We present a new MCMC algorithm for Bayesian adaptive Metropolis-Hasting sampling (BAMS), using an explicitly Bayesian inference to update the proposal distribution. We show that author Keith’s earlier Bayesian adaptive independence sampler (BAIS) and a new Bayesian adaptive random walk sampler (BARS) emerge as instances. More important than either of these instances, BAMS provides a general framework within which to explore adaptive schemes that are guaranteed to converge to the required limiting distribution.

In this article, we study the problem of estimating the stress-strength reliability, where the stress and strength variables follow independent exponential distributions with a common location parameter but different scale parameters. All parameters are assumed to be unknown. We derive the MLE, the UMVUE of the reliability parameter. We also derive the Bayes estimators considering conjugate prior distributions for the scale parameters and a dependent prior for the common location parameter. Monte Carlo simulations have been carried out to compare among the proposed estimators with respect to different loss functions.

Since in many life-testing situations it is not unlikely to note the unpredictable fluctuation of the scale parameter in a failure model, it is justifiable to consider such a parameter as a random variable and, thus, appeal to a Bayesian analysis. In specific, let 0 denote the random variable associated with the scale parameter and 0 its realization. Obviously, a Bayesian analysis depends on the utilization of prior information which, in this case, we assume to exist either in the form of a prior distribution of 0 or a sequence of sufficient statistics from past experiments. For the ordinary Bayes approach, we appeal to a well-known transformation to generalize the results of Bhattacharya (1967) for the one-parameter exponential model so as to include the flexibility provided by the shape parameter ? in the Weibull distribution. In fact, the Weibull failure model has an increasing ( > 1) or decreasing ( < 1) failure rate and, thus, is likely to describe the life-span of items with variable failure rates. The empirical Bayes estimation technique was largely motivated by Robbins (1955), who assumed the existence of a prior distribution for an unknown parameter but not the knowledge of its form. Instead, he substitutes past information which he assumes to exist as a result of the repetitive nature in the problem of estimation. Thus, in the absence of knowledge concerning the form of the prior distribution, we appeal to an empirical Bayes approach to estimate the scale parameter. By using this estimate, an estimate of the reliability function is made possible.

Multivariate random effects meta-analysis (MRMA) is an appropriate way for synthesizing data from studies reporting multiple correlated outcomes. In a Bayesian framework, it has great potential for integrating evidence from a variety of sources. In this paper, we propose a Bayesian model for MRMA of mixed outcomes, which extends previously developed bivariate models to the trivariate case and also allows for combination of multiple outcomes that are both continuous and binary. We have constructed informative prior distributions for the correlations by using external evidence. Prior distributions for the within-study correlations were constructed by employing external individual patent data and using a double bootstrap method to obtain the correlations between mixed outcomes. The between-study model of MRMA was parameterized in the form of a product of a series of univariate conditional normal distributions. This allowed us to place explicit prior distributions on the between-study correlations, which were constructed using external summary data. Traditionally, independent ‘vague’ prior distributions are placed on all parameters of the model. In contrast to this approach, we constructed prior distributions for the between-study model parameters in a way that takes into account the inter-relationship between them. This is a flexible method that can be extended to incorporate mixed outcomes other than continuous and binary and beyond the trivariate case. We have applied this model to a motivating example in rheumatoid arthritis with the aim of incorporating all available evidence in the synthesis and potentially reducing uncertainty around the estimate of interest. © 2013 The Authors. Statistics inMedicine Published by John Wiley & Sons, Ltd.

Estimating phylogenies in which branch lengths are expressed in units of absolute time is crucial for testing hypotheses in evolutionary biology. However, bioinformatic tools to pre- and post-process data from Bayesian divergence time estimation analyses are often not easily interoperable, and documenting methodological choices is not a generalized practice. The R package tbea is a tool-set to integrate biological, geological, and paleontological information to optimize the specification of models, their parameters, and prior distributions in divergence time estimation analyses. tbea implements statistical models to (i) better translate time information in dating sources into the specified calibration densities, (ii) improve comparisons between prior and posterior distributions for parameters of interest, (iii) carry out inference on origination times for a set of distributions, (iv) summarize different distributions into a single one, and (v) improve the reproducibility of divergence time estimation analyses allowing users to document methodological choices. We illustrate the functionalities of tbea by carrying out two worked examples, one on the phylogenetic relationships and divergence time estimation of South American Cynodontidae, and the other on the separation time of drainages east and west of the Andes in South America. It is expected that the tools available in tbea will be key when estimating events in time from sets of point estimates, and the combination of different posterior densities from the same parameter will be useful in justifying the selection of secondary calibration points, or discussing the timing of biogeographical events when multiple sources are available.

The Generalized Pareto (GP) and Generalized extreme value (GEV) distributions play an important role in extreme value analyses, as models for threshold excesses and block maxima respectively. For each of these distributions we consider Bayesian inference using "reference" prior distributions (in the general sense of priors constructed using formal rules) for the model parameters, specifically a Jeffreys prior, the maximal data information (MDI) prior and independent uniform priors on separate model parameters. We investigate the important issue of whether these improper priors lead to proper posterior distributions. We show that, in the GP and GEV cases, the MDI prior, unless modified, never yields a proper posterior and that in the GEV case this also applies to the Jeffreys prior. We also show that a sample size of three (four) is sufficient for independent uniform priors to yield a proper posterior distribution in the GP (GEV) case.