A robust Bayesian approach to modeling epistemic uncertainty in common-cause failure models

Abstract

In a standard Bayesian approach to the alpha-factor model for common-cause failure, a precise Dirichlet prior distribution models epistemic uncertainty in the alpha-factors. This Dirichlet prior is then updated with observed data to obtain a posterior distribution, which forms the basis for further inferences.In this paper, we adapt the imprecise Dirichlet model of Walley to represent epistemic uncertainty in the alpha-factors. In this approach, epistemic uncertainty is expressed more cautiously via lower and upper expectations for each alpha-factor, along with a learning parameter which determines how quickly the model learns from observed data. For this application, we focus on elicitation of the learning parameter, and find that values in the range of 1 to 10 seem reasonable. The approach is compared with Kelly and Atwood's minimally informative Dirichlet prior for the alpha-factor model, which incorporated precise mean values for the alpha-factors, but which was otherwise quite diffuse.Next, we explore the use of a set of Gamma priors to model epistemic uncertainty in the marginal failure rate, expressed via a lower and upper expectation for this rate, again along with a learning parameter. As zero counts are generally less of an issue here, we find that the choice of this learning parameter is less crucial.Finally, we demonstrate how both epistemic uncertainty models can be combined to arrive at lower and upper expectations for all common-cause failure rates. Thereby, we effectively provide a full sensitivity analysis of common-cause failure rates, properly reflecting epistemic uncertainty of the analyst on all levels of the common-cause failure model.

Predictive inference for system reliability after common-cause component failures

“…It is possible to select values α j based on background knowledge of the system, but in this paper we aim at learning about these probabilities from available failure data whilst attempting to add only rather minimal further assumptions. Recently, two Bayesian solutions to the same problem have been presented, one using a non-informative Dirichlet prior distribution [4] and the other using the imprecise Dirichlet model (IDM) [5]. Both of these let the numbers 1 to m of possible simultaneous failures be represented by m categories, with an assumed multinomial distribution for the numbers of observations in these categories.…”

Constructing copulas from shock models with imprecise distributions

“…For an exhaustive review on imprecise reliability, the reader is referred to [64]. In [63], an approach to common-cause failure models with imprecise probabilities is proposed. It is based on an idea that is somewhat similar to our shock models.…”

Sets of Priors Reflecting Prior-Data Conflict and Agreement

“…[11] suggested a parameter set shape that balances tractability and ease of elicitation with desired inference properties. This approach has been applied in common-cause failure modelling [7] and system reliability [12]. We further refine this approach by complementing the increased imprecision reaction to prior-data conflict with further reduced imprecision if prior and data coincide especially well, which we call strong prior-data agreement.…”