Discussion of Specifying prior distributions in reliability applications

Abstract

Congratulations on this great and comprehensive achievement. Undoubtedly, Bayesian inference plays an increasingly important role in reliability data analysis, dictated on the one hand by the usually small sample sizes per experimental condition, which bring standard frequentist procedures to their limits, and on the other hand by the fact that uncertainty quantification and communication are more straightforward in a Bayesian setup. Reliability data are mostly censored, with many realistic censoring schemes leading to complicated likelihood functions and posterior distributions that can be only approximated numerically with Markov Chain Monte Carlo (MCMC) methods. With the advances in Bayesian computation techniques and algorithms, this is however not a limitation anymore. The authors managed in this enlightening work to embed the reliability perspective view, grounded on the practitioners' needs, in a Bayesian theoretic setup, providing and commenting fundamental literature from both fields. This paper will be a valuable reference for practitioning Bayesian inference in reliability applications and, most importantly, for understanding the effect of the priors' choice. The provided insight on the role of a sensitivity analysis for the prior distribution is very important as well, especially when extrapolating results. Furthermore, the technical details and hints on the implementation in R will be highly appreciated.

It is not surprising, but good to see, that the essential role of the independence Jeffreys (IJ) priors is verified also in this context, for example, in cases of Type-I censoring with few observed failures. A crucial statement of the paper I would like to highlight is that in case of limited observed data, the usually “safe” choice of a noninformative prior can deliver misleading conclusions, since it may consider unlikely or impossible parts of the parameter space with high probability. Therefore, in reliability applications weakly informative priors that reflect the underlying framework or known effect of experimental conditions have to be prioritized. Moreover, along these lines, in case of experiments combining more than one experimental condition, if the level of the experimental condition has a monotone effect on the quantity of interest, say the expected lifetime, the choice of the priors under the different conditions should reflect this ordering. This is a direction of future research on Bayesian procedures for reliability applications with high expected impact.

In a Bayesian inferential framework, the derivation and use of credible intervals (CIs) is more natural and flexible than frequentist confidence intervals. In this work the focus lies on equal tailed CIs. For highly skewed posteriors, it would be of interest to consider in the future highest posterior density (HPD) CIs as well.

Motivated by the reference of the authors to Reference 1 and the priors in the framework of accelerated life testing (ALT), I would like to emphasize the importance of the choice of priors in ALT experiments and the associated challenges linked to the underlying type of censoring. Reference 1 considered noninformative priors for constant stress ALT (CSALT) experiments under Type-II censoring for Weibull lifetimes and studied their properties. For Weibull lifetimes but for step-stress ALT (SSALT) and a more complex censoring scheme (progressive interval censoring),² discussed simple conjugate-like priors along with normal priors, vague or noninformative, and implemented them on a simulated and a real example. For further sources on the Bayesian analysis of SSALT models, which was first dealt by Reference 3, we refer to Reference 2 and the references cited therein.

Furthermore, in CSALT and SSALT models with Weibull lifetimes it is usually assumed that the shape parameter $\beta$ is common across all $s$ considered stress levels and the stress change affects only the scale parameters ${\eta }_i$, $i=1,\dots ,s$. However, a stress increase may affect the shape of the lifetime distribution. For example, for a SSALT model with Weibull lifetimes⁴ considered the shape parameters to be increasing in the stress level, providing an associated physical justification, while² discussed on a real SSALT example with $s=4$ the impact on the predicted lifetime under normal operating condition of the assumption of a common $\beta$ or not. Thus, it is crucial to develop Bayesian procedures for deciding on the validity of a common $\beta$ assumption.

Although it is common evidence that Bayesian inference can provide an efficient solution to many complicated reliability applications, the obstacle for its broader use is the difficulty in adjusting and implementing Bayesian procedures in practice. For this, there is a demand for a user friendly package, tailored to the needs of reliability applications, that will provide a safe environment for Bayesian analysis without requiring Bayesian expertise. Under the functionalities of such a package should also be the possibility to transform practitioners' friendly specified prior information into adequate prior distributions.