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- Bayes statistics (1)
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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.
A fast quantitative model for the energy spectra of radiation emitted by X-ray tubes is described, handling thin as well as thick plane targets of arbitrary materials. The developed model is based on fundamental interaction cross sections, describing electron and photon transport via numerical integration of discretized distributions. While the focus lies on bremsstrahlung production, modeling of characteristic radiation is fully integrated. The model does not include any free parameters. The validity of the model is shown through comparison with measurements and Monte Carlo simulations for several combinations of target material and acceleration potential between 30 kV and 450 kV.