Discussion of “Specifying prior distributions in reliability applications”

Abstract

The authors should be congratulated on a very interesting and insightful discussion which motivates the use of Bayesian inference in reliability theory. The article motivates the use of Bayesian methods especially in the case of small number of failures. The following log-location-scale distributions are considered: the lognormal distribution and the Weibull distribution. The importance of reparameterization is discussed, where it is, for example, more useful to replace the scale parameter with a certain quantile. A very important advantage and practical reason for this is given as follows: “Elicitation of a prior distribution is facilitated because the parameter have practical interpretations and are familiar to practitioners.”

As stated in Irony and Singpurwalla,¹ José Bernardo said the following: “Non-subjective Bayesian analysis is just a part,—an important part, I believe-, of a healthy sensitivity analysis to the prior choice: it provides an answer to a very important question in scientific communication, namely, what could one conclude from the data if prior beliefs were such that the posterior distribution of the quantity of interest were dominated by the data.”

It would be interesting to see how a divergence prior will compare with the priors discussed in this article. Ghosh et al.² developed a prior where the distance between the prior and posterior is maximized by making use of the chi-square divergence, whereas a reference prior is the prior distribution that maximizes the Kullback–Leibler divergence between the prior and the posterior distribution. When other distances are used the Jeffreys prior is the result with adequate first order approximations but with the chi-square distance the second order approximations give this prior. Second order approximations is used since chi-square divergence approximations of the first order does not give priors. In other cases where other divergence measures are used, first order approximations gives priors that are adequate.

A distinction between weakly informative priors and noninformative priors is also given. A weakly informative prior as opposed to a noninformative prior is used when the prior influences the posterior mildly as opposed to having no influence on the posterior. The authors provide a very useful table of recommended prior distributions for log-location-scale distribution parameters, where it is clearly given and discussed what type of prior (informative, noninformative, or informative weakly informative) and prior distribution inputs are needed. A simulation study is done to investigate the coverage probability. When using complete data and Type 2 censored data from log-location-scale distribution, the independence Jeffreys prior have coverage rates that are the same as the nominal confidence level, and when using Type 2 and random censoring the independence Jeffreys prior have coverage rates that are close to the nominal confidence level. The article concludes with a sensitivity analysis, where different weakly informative priors are compared.

Throughout the article the applications and usefulness of the method/models are illustrated by using two data sets, the bearing cage field data from Abernethy et al.³ and the rocket motor field data from Olwell and Sorell.⁴