Elicitation of Weibull priors

Based on expert opinions, informative prior elicitation for the common Weibull lifetime distribution usually presents some difficulties since it requires to elicit a two-dimensional joint prior. We consider here a reliability framework where the available expert information states directly in terms of prior predictive values (lifetimes) and not parameter values, which are less intuitive. The novelty of our procedure is to weigh the expert information by the size m of a virtual sample yielding a similar information, the prior being seen as a reference posterior. Thus, the prior calibration by the Bayesian analyst, who has to moderate the subjective information with respect to the data information, is made simple. A main result is the full tractability of the prior under mild conditions, despite the conjugation issues encountered with the Weibull distribution. Besides, m is a practical focus point for discussion between analysts and experts, and a helpful parameter for leading sensitivity studies and reducing the potential imbalance in posterior selection between Bayesian Weibull models, which can be due to favoring arbitrarily a prior. The calibration of m is discussed and a real example is treated along the paper.

💡 Research Summary

The paper addresses the long‑standing difficulty of constructing informative joint priors for the Weibull lifetime distribution, whose two parameters (scale η and shape β) are hard for non‑statistical experts to assess directly. Instead of eliciting η and β, the authors propose to work with expert‑provided prior predictive information—specifically, quantiles of the lifetime distribution. An expert is asked to give a pair (tα, α), where tα is the lifetime such that the expert believes the probability of failure before tα equals α. This information is interpreted as the result of a virtual i.i.d. sample of size m, where m quantifies the amount of subjective information relative to the observed data.

The baseline non‑informative prior is taken as Jeffreys’ prior, πJ(η,β) ∝ η⁻¹ 1{η≥0} 1{β≥β0}, which is invariant to re‑parameterisation. Combining this with the likelihood of the virtual sample yields a tractable joint prior: conditional on β, η follows a Generalized Inverse‑Gamma (GIG) distribution, while β follows a truncated Gamma distribution. The sufficient statistics of the virtual sample, b(˜t_m,β)=∑˜t_i^β and β(˜t_m)=m/(∑log˜t_i), cannot be obtained directly from experts. The authors therefore replace b by a deterministic function bα(m,β)=((1−α)^{−1/m}−1) tα^β, which is the unique continuous mapping that guarantees the expert’s quantile condition Pπ(T<tα)=α.

The resulting prior is fully specified by two hyper‑parameters: the virtual sample size m and a shape‑parameter eβ(m) governing the Gamma prior for β. Calibration of eβ(m) can be performed using additional expert statements, such as the probability that the system exhibits aging (i.e., β exceeds a certain value), or by matching several credibility intervals { (tαi,αi) } supplied by the expert. The authors adopt a discrete Kullback‑Leibler loss between the desired predictive distribution (exactly satisfying the supplied intervals) and the predictive distribution induced by the candidate prior, and minimise this loss to obtain a unique eβ(m).

Once calibrated, the prior can be updated with real lifetime data t1,…,tn. Because η|β is GIG, Gibbs sampling is straightforward: draw β from its truncated Gamma full conditional, then draw η from the corresponding GIG. The approach thus yields a proper, analytically tractable prior that respects the expert’s predictive statements, automatically incorporates the η–β correlation, and provides an intuitive measure of information content via m.

A real‑world case study on secondary water‑circuit components in French nuclear plants illustrates the method. Two experts supplied different credibility intervals; the common median was adopted as the most trustworthy specification (MTS). Calibration yielded m≈3.5 and an appropriate eβ(m). Posterior summaries matched observed failure times, and a sensitivity analysis varying m demonstrated how increasing the virtual sample size strengthens the influence of the prior relative to the data—offering a clear, quantitative way for analysts and decision‑makers to discuss the balance between subjective and objective information.

The paper concludes that the virtual‑data‑based prior elicitation is not limited to Weibull models; any situation where experts can provide predictive quantiles rather than parameter values can benefit. Future work may extend the framework to multivariate lifetime models, hierarchical structures, and correlated expert opinions, further enhancing the defensibility and transparency of Bayesian reliability analyses.