Modelling and eliciting expert knowledge with fictitious data
The Bayesian approach is the optimal approach in statistical inference when prior knowledge from experts is available. Moreover, in such a context, the superiority of Bayesian inference over maximum likelihood can be important for small sample sizes. But, a reliable Bayesian analysis needs to take into account properly the prior knowledge. Usually, expressing expert opinions as central value of some parameters is not too difficult. But expressing the analyst doubt on expert opinions is more critical. In this communication, we are interested with this particular problem and we propose an approach where experts opinions are supposed to come from fictitious data. Acting in such a way allows the analyst to weight the importance of the expert opinion in regard to the actual sample size. For instance, it is often said that the expert opinion should not have more importance than the observed data. Using the strategy we now propose, it is possible to respect this constraint in a sensible and reliable way. The communication will be organized as follows. Firstly, since failure data sets are often censored, we present a simple way to measure the Fisher information derived from censored data. From that point, it becomes easy to measure the Fisher information brought by a possibly censored data set. Secondly, we consider a simple situation where a non informative Jeffreys prior distribution can be expressed as a conjugate prior distribution. And, we show that it is possible to express the expert prior knowledge as coming from a fictitious data set. It is then easy to calibrate the prior distribution to ensure that the expert contribution does not exceed the information brought by the observed sample. Some illustrations using standard lifetime distributions will be given. Third, we generalize this approach to a more general context where there is no possibility to use conjugate prior distributions, as it is for instance the case with models involving the Weibull distribution. This generalization is as follows. It is considered that the expert is able to derive a consistent (asymptotically unbiased) estimator with minimum variance. Under this assumption, the variance of the prior distribution parameter can be interpreted in terms of the Fisher information of the fictitious sample. It is then possible to weight the relative importance of the fictitious and the actual samples.