Bayesian reliability analysis with imprecise prior probabilities

Abstract

The Bayesian framework for statistical inference offers the possibility of taking expert opinions into account, and is therefore attractive in practical problems concerning the reliability of technical systems. Probability is the only language in which uncertainty can be consistently expressed, and this requires the use of prior distributions for reporting expert opinions. In this paper an extension of the standard Bayesian approach based on the theory of imprecise probabilities and intervals of measures is developed. It is shown that this is necessary to take the nature of experts' knowledge into account. The application of this approach in reliability theory is outlined. The concept of imprecise probabilities allows us to accept a range of possible probabilities from an expert for events of interest and thus makes the elicitation of prior information simpler and clearer. The method also provides a consistent way for combining the opinions of several experts.

On the Use of Imprecise Probabilities in Reliability

“…Several models have been suggested explicitly based on imprecise probabilities; we briefly discuss a few below. In addition, there is a rich literature on robust statistics, both Bayesian 10 and frequentist 11 ; suitable models presented there can also be used within imprecise probability theory 25 . This is generally discussed by Walley 4 .…”

Interval Enclosures for Reliability Metrics

“…and m(A i ) are the width and the midpoint of A i , respectively, and F and G are interval extensions [20,21] for f (t ) and its 4-order derivative f (4) …”

“…These studies present reliability interval analysis aimed to determine failure probabilities with interval parameters. In [4] it is outlined a focus in providing intervals to reliability based on Bayesian analysis. The work of [16] presents interval enclosures for reliability function values of systems with Exponential failure distribution.…”

Decision Making with Imprecise Probabilities

“…be useful in reliability analysis, since quite often in reliability problems a lack of historical or experimental data forces us to use expert opinion. Coolen and Newby (1993) have introduced the above concept into this area, where often the variable of interest is lifetime of equipment, whereas suitable parametrical models such as the Weibull or lognormal distributions have parameters that are hard to interpret, so indeed it seems best to concentrate on elicitation methods through questions about the lifetime.…”