Discussion of: Specifying prior distributions in reliability applications

Abstract

This interesting paper by Tian et al. presents a comprehensive investigation of non-informative and weakly informative priors for two parameter (log-location and scale) failure distributions. They provide helpful and practical advice to the Bayesian analyst on the selection of appropriate priors and specifically on the avoidance of posterior estimates that are unrealistic, particularly where data are sparse.

The motivating examples provide challenging settings where the information provided by the data is extremely slight. These settings are typical of systems engineered to be very high reliable, where failure data are minimal by design, but where inferences about failure risk are critical. These are also precisely the settings where default choices for noninformative priors may be unexpectedly influential,¹ leading either to improper posteriors, or to posteriors which place significant mass in regions which are implausible. The authors' fundamental principle (§5.4) of ensuring that the priors always be constructed to avoid this consequence is very well stated, and one which will bear much repetition in other forums.

We have only one main point to make. It relates to their statement in the abstract that ‘for Bayesian inference, there is only one method of constructing equal-tailed credible intervals—but it is necesssary to provide a prior distribution to full specify the model.’ We agree, but our view is that the statement is incomplete: the model must have been chosen to begin with. Although this is not the main point of the paper, the consequences of model choice can be considerable, particularly when all of the inferential action is being carried out on the tails of the distribution, where only a few percent of failures may ever be observed to occur.

In this spirit we have reproduced in our Figure 1 the authors' Weibull probability plot (their Figure 1) of the Bearing Cage failure data.² The estimated parameters of the original Weibull fit are $(\hat{\beta },\hat{\eta })=(2.035,11792)$, and we show that fitted cumulative distribution function with a solid line. Choosing the tail areas $p=0.1$ and $q=0.005$ these correspond to the parameter values $({\hat{t}}_p,{\hat{\lambda }}_q)=(3902,0.224)$, and we have drawn horizontal dashed lines corresponding to $p$ and $q$. A vertical line at $t=8000$ h marks a key point of inferential interest. We have also shown cumulative distributions of a log Normal and a Gamma distribution with matching values of $(t_p,{\lambda }_q)$.

The tail shapes do of course differ somewhat from one another, the log Normal most markedly. Inference about the 8000 hour point is necessarily affected not only by the choice of prior but also by the choice of model.

A paper discussing model selection would have been a rather different one than the current work by Tian et al. Nevertheless we raise the question of whether, if the focus is predominantly on the lower tail of the distribution, a suitable choice of prior on top of a single assumed likelihood be able to do some of the same heavy lifting as a prior on the space of models. Moreover, a prior on $(t_p,{\lambda }_q)$ might yield further advantages in elicitation – given that the large scale parameter $\sigma$ is presumably harder to characterise then a second quantile $q$ would be. Even if this proves difficult, future work might helpfully incorporate comments and guidance on model selection.

In conclusion, we thank the authors for their comprehensive treatment of the question of prior specification, and the practical guidance they provide.