Discussion specifying prior distributions in reliability applications

Abstract

Firstly, I want to congratulate the authors in Reference 1 for their practical contextualization in describing the Bayesian method in real-world problems with reliability data. Undoubtedly, one of the main strengths of this article is its highly practical approach, starting from real situations and examples, and showing why Bayesian inference is many times a nice alternative for making estimations. The authors nicely describe how, in reliability applications, there are generally few failure records and, therefore, little information available. For example, this is often the case in the study of the reliability of engineering systems in the army, such as some types of weapons. Since the specific prior is a key aspect of the Bayesian framework, they are primarily concerned with guiding readers on how to make this choice properly.

Once the parameter of interest $\theta =(\mu ,\sigma )$ has been identified, and without losing sight of real-world applications, the authors develop their exposition based on three essential premises. Firstly, they remind us that the distribution of $\theta$ may not always be the main focus of our interest in practical situations. Instead, our key objective might involve estimating cumulative failure probabilities at a specific time or a failure-time distribution $p$-quantile, given by the expression $t_p=\exp [\mu +{\Phi }^{-1}(p\left)\sigma \right]$, where $p\in (0,1)$. Secondly, censored data underly the essence of reliability analysis. Therefore, right, interval, and left censored observations play a fundamental role in all our estimations. Lastly, the authors emphasize that certain reparameterizations of the parameter $\theta$ can sometimes facilitate the practical interpretation of new parameters and enable greater mathematical tractability. For instance, replacing the usual scale parameter $\exp \left(\mu \right)$ with a specific quantile $t_p$ can be useful in practice. Building upon these three considerations, the authors exhaustively describe the most commonly used techniques for eliciting a prior distribution and provide a substantial number of bibliographic citations. This fact is valuable in itself because it enables the reader to be aware of the real issues associated with their data and the various approaches available to address the estimation problem.

One of the most positive aspects of this work is the effort made by the authors to describe most of the known procedures for choosing the prior distribution for the log-location-scale family. The authors provide a summary of the state of the art concerning the elicitation of non-informative distributions, informative distributions, expert opinions, or a combination of various techniques. Specifically, several methods for choosing non-informative priors are comprehensively presented, such as a Jeffreys prior, which is proportional to the square root of the determinant of the Fisher information matrix (FIM), an independence Jeffreys (IJ) prior based on the Conditional Jeffreys (CJ) prior for each parameter, a reference prior that maximizes the Kullback–Leibler divergence between the prior and the expected posterior distribution, and an ordered reference prior that specifies the order of importance of the parameter. The authors also describe the relationships among these non-informative priors, showing when some have advantages over others depending on the nature of the data. In this regard, one of the most useful contributions made by the authors is the elaboration of Table 1 in their article, which summarizes the Jeffreys, IJ, and reference non-informative prior distributions for the log-location-scale family using different parameterizations and censoring scenarios. It is also remarkable Table 2 in their article, where the authors provide a summary of the recommended prior distributions for use with log-location-scale distributions.

In my opinion, a detailed discussion about the choice of the prior distribution in other complex models could improve the article. For example, in failure processes of heterogeneous repairable systems, which are often modeled by non-homogeneous Poisson processes (NHPP). In these processes, we observe the total number of failures that occur in an engineering system in an interval time $(0,t]$, and the estimation of the parameters of the intensity function of the model, $\lambda \left(t\right)$, is crucial. It that sense, the authors explicitly mention the work described in Reference 2, where it is argued for the use of prior information for $\lambda \left(t\right)$. Other interesting articles in this regard are References 3 and 4, where the authors postulate different forms of the intensity function in a study in Bayesian reliability analysis concerning train door failures on a European underground system. The problem in this context is twofold. First, choosing the intensity function, and secondly, how to evaluate the prior information about its parameters. For example, if the Weibull distribution governs the first system failure, it leads us to the popular power law process (PLP) with intensity function $\lambda \left(t\right|\theta )=M\beta t^{\beta -1}$, $\theta =(M,\beta )\in {ℝ}^{+}\times {ℝ}^{+}$. The choice of prior distributions for $M$ and $\beta$ and its connection with the results presented in Tables 1 and 2, in their article, deserve an in-detph study.

Regarding other interesting models, I would also like to point out about potential applications in Metrology. The role of Metrology in engineering is essential because it ensures the functionality of measuring equipment, proper calibration, and quality control. Metrology suffers from the same problem as reliability data. Due to economic constraints there is a limitation of information where it is common having a random sample of size one, see⁵ for a recent application of Bayesian techniques to evaluate measurement data.

With respect to the sensitivity, the authors are aware that the choice of a prior distribution could have a strong influence on inferences when having limited information in the data. Therefore, they address in their Section 10 a parametric analysis of sensitivity. At this respect, I would like to pay attention to recent articles about the robustness of Bayesian methods which have potential applications in reliability. All these new methods can improve the robustness study.

To conclude my discussion, I do think the authors have nicely summarized most of the known methods to choice a specific prior distribution in the log-location-scale family of distributions. As a clear strength of the article, all methods are adjusted to the most practical realities based on censoring scheme. Finally, the exhaustive overview described by the authors opens new problems in other complex model as in the NHPP one. Additionally, the properties of the bivariate likelihood function could lead us to a more precise analysis of sensitivity.