Comparison of Bayesian models for production efficiency

In this paper, we use Markov Chain Monte Carlo (MCMC) methods in order to estimate and compare stochastic production frontier models from a Bayesian perspective. We consider a number of competing models in terms of different production functions and the distribution of the asymmetric error term. All MCMC simulations are done using the package JAGS (Just Another Gibbs Sampler), a clone of the classic BUGS package which works closely with the R package where all the statistical computations and graphics are done.     

Bayesian analysis of generalized elliptical semi-parametric models

In this paper, we study the statistical inference based on the Bayesian approach for regression models with the assumption that independent additive errors follow normal, Student-t, slash, contaminated normal, Laplace or symmetric hyperbolic distribution, where both location and dispersion parameters of the response variable distribution include nonparametric additive components approximated by B-splines. This class of models provides a rich set of symmetric distributions for the model error. Some of these distributions have heavier or lighter tails than the normal as well as different levels of kurtosis. In order to draw samples of the posterior distribution of the interest parameters, we propose an efficient Markov Chain Monte Carlo (MCMC) algorithm, which combines Gibbs sampler and Metropolis–Hastings algorithms. The performance of the proposed MCMC algorithm is assessed through simulation experiments. We apply the proposed methodology to a real data set. The proposed methodology is implemented in the R package BayesGESM using the function gesm().     

A Hellinger distance approach to MCMC diagnostics

Bayesian analysis often requires the researcher to employ Markov Chain Monte Carlo (MCMC) techniques to draw samples from a posterior distribution which in turn is used to make inferences. Currently, several approaches to determine convergence of the chain as well as sensitivities of the resulting inferences have been developed. This work develops a Hellinger distance approach to MCMC diagnostics. An approximation to the Hellinger distance between two distributions f and g based on sampling is introduced. This approximation is studied via simulation to determine the accuracy. A criterion for using this Hellinger distance for determining chain convergence is proposed as well as a criterion for sensitivity studies. These criteria are illustrated using a dataset concerning the Anguilla australis, an eel native to New Zealand.     

Estimation of the Burr type III distribution with application in unified hybrid censored sample of fracture toughness

In this paper, the statistical inference of the unknown parameters of a Burr Type III (BIII) distribution based on the unified hybrid censored sample is studied. The maximum likelihood estimators of the unknown parameters are obtained using the Expectation–Maximization algorithm. It is observed that the Bayes estimators cannot be obtained in explicit forms, hence Lindley's approximation and the Markov Chain Monte Carlo (MCMC) technique are used to compute the Bayes estimators. Further the highest posterior density credible intervals of the unknown parameters based on the MCMC samples are provided. The new model selection test is developed in discriminating between two competing models under unified hybrid censoring scheme. Finally, the potentiality of the BIII distribution to analyze the real data is illustrated by using the fracture toughness data of the three different materials namely silicon nitride (Si₃N₄), Zirconium dioxide (ZrO₂) and sialon (Si_6?xAl_xO_xN_8?x). It is observed that for the present data sets, the BIII distribution has the better fit than the Weibull distribution which is frequently used in the fracture toughness data analysis.     

Marginal maximum a posteriori estimation using Markov chain Monte Carlo

Markov chain Monte Carlo (MCMC) methods, while facilitating the solution of many complex problems in Bayesian inference, are not currently well adapted to the problem of marginal maximum a posteriori (MMAP) estimation, especially when the number of parameters is large. We present here a simple and novel MCMC strategy, called State-Augmentation for Marginal Estimation (SAME), which leads to MMAP estimates for Bayesian models. We illustrate the simplicity and utility of the approach for missing data interpolation in autoregressive time series and blind deconvolution of impulsive processes.     

Multivariate process capability a bayesian perspective

In this paper an attempt has been made to examine the multivariate versions of the common process capability indices (PCI's) denoted by C_p and C_pk . Markov chain Monte Carlo (MCMC) methods are used to generate sampling distributions for the various PCI's from where inference is performed. Some Bayesian model checking techniques are developed and implemented to examine how well our model fits the data. Finally the methods are exemplified on a historical aircraft data set collected by the Pratt and Whitney Company.     

Inference for Adaptive Time Series Models: Stochastic Volatility and Conditionally Gaussian State Space Form

In this paper we model the Gaussian errors in the standard Gaussian linear state space model as stochastic volatility processes. We show that conventional MCMC algorithms for this class of models are ineffective, but that the problem can be alleviated by reparameterizing the model. Instead of sampling the unobserved variance series directly, we sample in the space of the disturbances, which proves to lower correlation in the sampler and thus increases the quality of the Markov chain.

Using our reparameterized MCMC sampler, it is possible to estimate an unobserved factor model for exchange rates between a group of n countries. The underlying n + 1 country-specific currency strength factors and the n + 1 currency volatility factors can be extracted using the new methodology. With the factors, a more detailed image of the events around the 1992 EMS crisis is obtained.

We assess the fit of competitive models on the panels of exchange rates with an effective particle filter and find that indeed the factor model is strongly preferred by the data.     

Markov chain Monte Carlo with the Integrated Nested Laplace Approximation

The Integrated Nested Laplace Approximation (INLA) has established itself as a widely used method for approximate inference on Bayesian hierarchical models which can be represented as a latent Gaussian model (LGM). INLA is based on producing an accurate approximation to the posterior marginal distributions of the parameters in the model and some other quantities of interest by using repeated approximations to intermediate distributions and integrals that appear in the computation of the posterior marginals. INLA focuses on models whose latent effects are a Gaussian Markov random field. For this reason, we have explored alternative ways of expanding the number of possible models that can be fitted using the INLA methodology. In this paper, we present a novel approach that combines INLA and Markov chain Monte Carlo (MCMC). The aim is to consider a wider range of models that can be fitted with INLA only when some of the parameters of the model have been fixed. We show how new values of these parameters can be drawn from their posterior by using conditional models fitted with INLA and standard MCMC algorithms, such as Metropolis–Hastings. Hence, this will extend the use of INLA to fit models that can be expressed as a conditional LGM. Also, this new approach can be used to build simpler MCMC samplers for complex models as it allows sampling only on a limited number of parameters in the model. We will demonstrate how our approach can extend the class of models that could benefit from INLA, and how the R-INLA package will ease its implementation. We will go through simple examples of this new approach before we discuss more advanced applications with datasets taken from the relevant literature. In particular, INLA within MCMC will be used to fit models with Laplace priors in a Bayesian Lasso model, imputation of missing covariates in linear models, fitting spatial econometrics models with complex nonlinear terms in the linear predictor and classification of data with mixture models. Furthermore, in some of the examples we could exploit INLA within MCMC to make joint inference on an ensemble of model parameters.     

Extended Weibull type distribution and finite mixture of distributions

An extended form of Weibull distribution is suggested which has two shape parameters (m and δ). Introduction of another shape parameter δ helps to express the extended Weibull distribution not only as an exact form of a mixture of distributions under certain conditions, but also provides extra flexibility to the density function over positive range. The shape of density function of the extended Weibull type distribution for various values of the parameters is shown which may be of some interest to Bayesians. Certain statistical properties such as hazard rate function, mean residual function, rth moment are defined explicitly. The proposed extended Weibull distribution is used to derive an exact form of two, three and k-component mixture of distributions. With the help of a real data set, the usefulness of mixture Weibull type distribution is illustrated by using Markov Chain Monte Carlo (MCMC), Gibbs sampling approach.     

BINARY REGRESSION WITH A CLASS OF SKEWED t LINK MODELS

ABSTRACT

In this paper we propose a class of skewed t link models for analyzing binary response data with covariates. It is a class of asymmetric link models designed to improve the overall fit when commonly used symmetric links, such as the logit and probit links, do not provide the best fit available for a given binary response dataset. Introducing a skewed t distribution for the underlying latent variable, we develop the class of models. For the analysis of the models, a Bayesian and non-Bayesian methods are pursued using a Markov chain Monte Carlo (MCMC) sampling based approach. Necessary theories involved in modelling and computation are provided. Finally, a simulation study and a real data example are used to illustrate the proposed methodology.     

Model fitting and inference under latent equilibrium processes

This paper presents a methodology for model fitting and inference in the context of Bayesian models of the type f(Y | X,θ)f(X|θ)f(θ), where Y is the (set of) observed data, θ is a set of model parameters and X is an unobserved (latent) stationary stochastic process induced by the first order transition model f(X ^(t+1)|X ^(t),θ), where X ^(t) denotes the state of the process at time (or generation) t. The crucial feature of the above type of model is that, given θ, the transition model f(X ^(t+1)|X ^(t),θ) is known but the distribution of the stochastic process in equilibrium, that is f(X|θ), is, except in very special cases, intractable, hence unknown. A further point to note is that the data Y has been assumed to be observed when the underlying process is in equilibrium. In other words, the data is not collected dynamically over time. We refer to such specification as a latent equilibrium process (LEP) model. It is motivated by problems in population genetics (though other applications are discussed), where it is of interest to learn about parameters such as mutation and migration rates and population sizes, given a sample of allele frequencies at one or more loci. In such problems it is natural to assume that the distribution of the observed allele frequencies depends on the true (unobserved) population allele frequencies, whereas the distribution of the true allele frequencies is only indirectly specified through a transition model. As a hierarchical specification, it is natural to fit the LEP within a Bayesian framework. Fitting such models is usually done via Markov chain Monte Carlo (MCMC). However, we demonstrate that, in the case of LEP models, implementation of MCMC is far from straightforward. The main contribution of this paper is to provide a methodology to implement MCMC for LEP models. We demonstrate our approach in population genetics problems with both simulated and real data sets. The resultant model fitting is computationally intensive and thus, we also discuss parallel implementation of the procedure in special cases.     

On population-based simulation for static inference

In this paper we present a review of population-based simulation for static inference problems. Such methods can be described as generating a collection of random variables {X _n } _n=1,…,N in parallel in order to simulate from some target density π (or potentially sequence of target densities). Population-based simulation is important as many challenging sampling problems in applied statistics cannot be dealt with successfully by conventional Markov chain Monte Carlo (MCMC) methods. We summarize population-based MCMC (Geyer, Computing Science and Statistics: The 23rd Symposium on the Interface, pp. 156–163, 1991; Liang and Wong, J. Am. Stat. Assoc. 96, 653–666, 2001) and sequential Monte Carlo samplers (SMC) (Del Moral, Doucet and Jasra, J. Roy. Stat. Soc. Ser. B 68, 411–436, 2006a), providing a comparison of the approaches. We give numerical examples from Bayesian mixture modelling (Richardson and Green, J. Roy. Stat. Soc. Ser. B 59, 731–792, 1997).     

A flexible statistical and efficient computational approach to object location applied to electrical tomography

In electrical tomography, multiple measurements of voltage are taken between electrodes on the boundary of a region with the aim of investigating the electrical conductivity distribution within the region. The relationship between conductivity and voltage is governed by an elliptic partial differential equation derived from Maxwell’s equations. Recent statistical approaches, combining Bayesian methods with Markov chain Monte Carlo (MCMC) algorithms, allow to greater flexibility than classical inverse solution approaches and require only the calculation of voltages from a conductivity distribution. However, solution of this forward problem still requires the use of the Finite Difference Method (FDM) or the Finite Element Method (FEM) and many thousands of forward solutions are needed which strains practical feasibility. Many tomographic applications involve locating the perimeter of some homogeneous conductivity objects embedded in a homogeneous background. It is possible to exploit this type of structure using the Boundary Element Method (BEM) to provide a computationally efficient alternative forward solution technique. A geometric model is then used to define the region boundary, with priors on boundary smoothness and on the range of feasible conductivity values. This paper investigates the use of a BEM/MCMC approach for electrical resistance tomography (ERT) data. The efficiency of the boundary-element method coupled with the flexibility of the MCMC technique gives a promising new approach to object identification in electrical tomography. Simulated ERT data are used to illustrate the procedures.     

Bayesian inference for pairwise interacting point processes

Pairwise interacting point processes are commonly used to model spatial point patterns. To perform inference, the established frequentist methods can produce good point estimates when the interaction in the data is moderate, but some methods may produce severely biased estimates when the interaction in strong. Furthermore, because the sampling distributions of the estimates are unclear, interval estimates are typically obtained by parametric bootstrap methods. In the current setting however, the behavior of such estimates is not well understood. In this article we propose Bayesian methods for obtaining inferences in pairwise interacting point processes. The requisite application of Markov chain Monte Carlo (MCMC) techniques is complicated by an intractable function of the parameters in the likelihood. The acceptance probability in a Metropolis-Hastings algorithm involves the ratio of two likelihoods evaluated at differing parameter values. The intractable functions do not cancel, and hence an intractable ratio r must be estimated within each iteration of a Metropolis-Hastings sampler. We propose the use of importance sampling techniques within MCMC to address this problem. While r may be estimated by other methods, these, in general, are not readily applied in a Bayesian setting. We demonstrate the validity of our importance sampling approach with a small simulation study. Finally, we analyze the Swedish pine sapling dataset (Strand 1972) and contrast the results with those in the literature.     

Diffusive nested sampling

We introduce a general Monte Carlo method based on Nested Sampling (NS), for sampling complex probability distributions and estimating the normalising constant. The method uses one or more particles, which explore a mixture of nested probability distributions, each successive distribution occupying ∼e ⁻¹ times the enclosed prior mass of the previous distribution. While NS technically requires independent generation of particles, Markov Chain Monte Carlo (MCMC) exploration fits naturally into this technique. We illustrate the new method on a test problem and find that it can achieve four times the accuracy of classic MCMC-based Nested Sampling, for the same computational effort; equivalent to a factor of 16 speedup. An additional benefit is that more samples and a more accurate evidence value can be obtained simply by continuing the run for longer, as in standard MCMC.     

Parallel tempering for dynamic generalized linear models

ABSTRACT

Markov chain Monte Carlo (MCMC) methods can be used for statistical inference. The methods are time-consuming due to time-vary. To resolve these problems, parallel tempering (PT), as a parallel MCMC method, is tried, for dynamic generalized linear models (DGLMs), as well as the several optimal properties of our proposed method. In PT, two or more samples are drawn at the same time, and samples can exchange information with each other. We also present some simulations of the DGLMs in the case and provide two applications of Poisson-type DGLMs in financial research.     

Classical and Bayesian estimation of stress-strength reliability in type II censored Pareto distributions

ABSTRACT

In this paper, the stress-strength reliability, R, is estimated in type II censored samples from Pareto distributions. The classical inference includes obtaining the maximum likelihood estimator, an exact confidence interval, and the confidence intervals based on Wald and signed log-likelihood ratio statistics. Bayesian inference includes obtaining Bayes estimator, equi-tailed credible interval, and highest posterior density (HPD) interval given both informative and non-informative prior distributions. Bayes estimator of R is obtained using four methods: Lindley's approximation, Tierney-Kadane method, Monte Carlo integration, and MCMC. Also, we compare the proposed methods by simulation study and provide a real example to illustrate them.     

Relabel mixture models via modal clustering

Effectively solving the label switching problem is critical for both Bayesian and Frequentist mixture model analyses. In this article, a new relabeling method is proposed by extending a recently developed modal clustering algorithm. First, the posterior distribution is estimated by a kernel density from permuted MCMC or bootstrap samples of parameters. Second, a modal EM algorithm is used to find the m! symmetric modes of the KDE. Finally, samples that ascend to the same mode are assigned the same label. Simulations and real data applications demonstrate that the new method provides more accurate estimates than many existing relabeling methods.     

On parallelizable Markov chain Monte Carlo algorithms with waste-recycling

Parallelizable Markov chain Monte Carlo (MCMC) generates multiple proposals and parallelizes the evaluations of the likelihood function on different cores at each MCMC iteration. Inspired by Calderhead (Proc Natl Acad Sci 111(49):17408–17413, 2014), we introduce a general ‘waste-recycling’ framework for parallelizable MCMC, under which we show that using weighted samples from waste-recycling is preferable to resampling in terms of both statistical and computational efficiencies. We also provide a simple-to-use criteria, the generalized effective sample size, for evaluating efficiencies of parallelizable MCMC algorithms, which applies to both the waste-recycling and the vanilla versions. A moment estimator of the generalized effective sample size is provided and shown to be reasonably accurate by simulations.     

Marginal Likelihood Estimation with the Cross-Entropy Method

We consider an adaptive importance sampling approach to estimating the marginal likelihood, a quantity that is fundamental in Bayesian model comparison and Bayesian model averaging. This approach is motivated by the difficulty of obtaining an accurate estimate through existing algorithms that use Markov chain Monte Carlo (MCMC) draws, where the draws are typically costly to obtain and highly correlated in high-dimensional settings. In contrast, we use the cross-entropy (CE) method, a versatile adaptive Monte Carlo algorithm originally developed for rare-event simulation. The main advantage of the importance sampling approach is that random samples can be obtained from some convenient density with little additional costs. As we are generating independent draws instead of correlated MCMC draws, the increase in simulation effort is much smaller should one wish to reduce the numerical standard error of the estimator. Moreover, the importance density derived via the CE method is grounded in information theory, and therefore, is in a well-defined sense optimal. We demonstrate the utility of the proposed approach by two empirical applications involving women's labor market participation and U.S. macroeconomic time series. In both applications, the proposed CE method compares favorably to existing estimators.