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 eliciting informative priors for the Weibull lifetime distribution, which traditionally requires specifying a joint prior over its two parameters (shape α and scale β). Because experts find it hard to express beliefs directly about these abstract parameters, the authors propose a framework that works with the quantities experts naturally understand: predictive lifetimes such as median life, mean time to failure, or quantiles at specified reliability levels.

The central innovation is to treat the expert’s predictive statements as if they originated from a “virtual sample” of size m. The virtual sample is assumed to follow the same Weibull model as the real data, and the expert’s reported lifetimes are taken as sufficient statistics of that sample. By raising the likelihood contributed by the expert’s k reported values to the power m/k, the prior density becomes

π(α,β | expert) ∝