Rare events – a probability approach or an ill-posed problem?

Rare events are understood to be events occurring once in a while but with dramatic consequences. Their occurrence cannot be predicted precisely, only a probability might be estimated, for example from past experiences. However, it might be rather misleading to attempt to describe them by distribution curves that apply for frequent or repeated observations, such as the Gaussian bell shape. Alternative distributions have been introduced to characterise the intervals at which a certain event may occur. It is the aim of technical safety and public security to prevent adverse events. Detectable indications that are typical for their course and are observable have to be identified before an incident occurs. Since they should be characteristic for such cases, they themselves also constitute rare events. The problem encountered in any detection system is that nothing is perfect. As in medical diagnostics, true indications may be missed or false test responses may pretend to be something that does not exist. Balancing missed indications with false positive calls is achieved with the aid of the so-called receiver operating characteristics (ROC). However, with the aid of Bayes' inference it can be shown that identifying signs of a rarely occurring indication is like looking for a needle in a haystack, even with an excellent detection approach with a low miss rate and an even lower probability of false calls. The inclusion of additionally available information may lead to a more effective search strategy. When employing imaging methods for detecting flaws or illicit items, the identification of rare indications can be impeded by blurring noise or overlapping items. The identification of the features sought can be supported by including information on their typical characteristics by regularisation algorithms. The strategy of such an approach is demonstrated in a simplified example with a plain geometric figure (circle) corrupted with structural noise. The shape of the original figure was clearly recovered. In general, search strategies should aim at an indication typical for the event to be prevented; otherwise, alternative approaches have to be considered, including, perhaps, serendipity.