M. J. Rufo

Sensitivity estimations for Bayesian inference models solved by MCMC methods

The advent of Markov Chain Monte Carlo (MCMC) methods to simulate posterior distributions has virtually revolutionized the practice of Bayesian statistics. Unfortunately, sensitivity analysis in MCMC methods is a difficult task. In this paper, a computationally low-cost method to estimate local parametric sensitivities in Bayesian models is proposed. The sensitivity measure considered here is the gradient vector of a posterior quantity with respect to the parameter. The gradient vector components are estimated by using a result based on the integral/derivative interchange. The MCMC simulations used to estimate the posterior quantity can be re-used to estimate the sensitivity measures and their errors, avoiding the need for further sampling. The proposed method is easy to apply in practice as it is shown with an illustrative example.

MCMC-based local parametric sensitivity estimations

Bayesian inferences for complex models need to be made by approximation techniques, mainly by Markov chain Monte Carlo (MCMC) methods. For these models, sensitivity analysis is a difficult task. A novel computationally low-cost approach to estimate local parametric sensitivities in Bayesian models is proposed. This method allows to estimate the sensitivity measures and their errors with the same random sample that has been generated to estimate the quantity of interest. Conditions to allow a derivative-integral interchange in the operator of interest are required. Two illustrative examples have been considered to show how sensitivity computations with respect to the prior distribution and the loss function are easily obtained in practice.

New approaches to compute Bayes factor in finite mixture models

Two new approaches to estimate Bayes factors in a finite mixture model context are proposed. Specifically, two algorithms to estimate them and their errors are derived by decomposing the resulting marginal densities. Then, through Bayes factor comparisons, the appropriate number of components for the mixture model is obtained. The approaches are based on simple theory (Monte Carlo methods and cluster sampling), what makes them appealing tools in this context. The performance of both algorithms is studied for different situations and the procedures are illustrated with some previously published data sets.

Local parametric sensitivity for mixture models of lifetime distributions

Mixture models are receiving considerable significance in the last years. Practical situations in reliability and survival analysis may be addressed by using mixture models. When making inferences on them, besides the estimates of the parameters, a sensitivity analysis is necessary. In this paper, a general technique to estimate local prior sensitivities in finite mixtures of distributions from natural exponential families having quadratic variance function (NEF-QVF) is proposed. Those families include some distributions of wide use in reliability theory. An advantage of this method is that it allows a direct implementation of the sensitivity measure estimates and their errors. In addition, the samples that are drawn to estimate the parameters in the mixture model are re-used to estimate the sensitivity measures and their errors. An illustrative application based on insulating fluid failure data is shown.

Bayesian analysis of finite mixtures of multinomial and negative-multinomial distributions

The Bayesian implementation of finite mixtures of distributions has been an area of considerable interest within the literature. Computational advances on approximation techniques such as Markov chain Monte Carlo (MCMC) methods have been a keystone to Bayesian analysis of mixture models. This paper deals with the Bayesian analysis of finite mixtures of two particular types of multidimensional distributions: the multinomial and the negative-multinomial ones. A unified framework addressing the main topics in a Bayesian analysis is developed for the case with a known number of component distributions. In particular, theoretical results and algorithms to solve the label-switching problem are provided. An illustrative example is presented to show that the proposed techniques are easily applied in practice.

Quasi-Random Sampling Importance Resampling

ABSTRACT We propose two modifications of the sampling/importance resampling (SIR) algorithm introduced by Rubin (1988). They are based on the use of low-discrepancy point sets and sequences. The proposed algorithms yield more representative samples in the sense of the F-discrepancy that turns into better estimations of summary inferences. Although no theoretical proof is provided, an empirical study through a wide range of distributions shows that the proposed approaches improve the SIR algorithm. We include some examples which are illustrative in this sense.

Adversarial life testing: A Bayesian negotiation model

Life testing is a procedure intended for facilitating the process of making decisions in the context of industrial reliability. On the other hand, negotiation is a process of making joint decisions that has one of its main foundations in decision theory. A Bayesian sequential model of negotiation in the context of adversarial life testing is proposed. This model considers a general setting for which a manufacturer offers a product batch to a consumer. It is assumed that the reliability of the product is measured in terms of its lifetime. Furthermore, both the manufacturer and the consumer have to use their own information with respect to the quality of the product. Under these assumptions, two situations can be analyzed. For both of them, the main aim is to accept or reject the product batch based on the product reliability. This topic is related to a reliability demonstration problem. The procedure is applied to a class of distributions that belong to the exponential family. Thus, a unified framework addressing the main topics in the considered Bayesian model is presented. An illustrative example shows that the proposed technique can be easily applied in practice.

Addressing misclassification for binary data: probit and t-link regressions

Generalized linear models are addressed to describe the dependence of data on explanatory variables when the binary outcome is subject to misclassification. Both probit and t-link regressions for misclassified binary data under Bayesian methodology are proposed. The computational difficulties have been avoided by using data augmentation. The idea of using a data augmentation framework (with two types of latent variables) is exploited to derive efficient Gibbs sampling and expectation–maximization algorithms. Besides, this formulation has allowed to obtain the probit model as a particular case of the t-link model. Simulation examples are presented to illustrate the model performance when comparing with standard methods that do not consider misclassification. In order to show the potential of the proposed approaches, a real data problem arising when studying hearing loss caused by exposure to occupational noise is analysed.

A Bayesian negotiation model for quality and price in a multi-consumer context

Bayesian decision theory plays a significant role in a large number of applications that have as main aim decision making. At the same time, negotiation is a process of making joint decisions that has one of its main foundations in decision theory. In this context, an important issue involved in industrial and commercial applications is product reliability/quality demonstration. The goal is, among others, product commercialization with the best possible price. This paper provides a Bayesian sequential negotiation model in the context of sale of a product based on two characteristics: product price and reliability/quality testing. The model assumes several parties, a manufacturer and different consumers, who could be considered adversaries. In addition, a general setting for which the manufacturer offers a product batch to the consumers is taken. Both the manufacturer and the consumers have to use their prior beliefs as well as their preferences. Sometimes, the model will require to update the previous beliefs. This can be made through the corresponding posterior distribution. Anyway, the main aim is that at least one consumer accepts the product batch based on either product price or product price and reliability/quality. The general model is solved from the manufacturer viewpoint. Thus a general approach that allows us to calculate an optimal price and sample size for testing is provided. Finally, two applications show how the proposed technique can be applied in practice.