A Bayesian Baseline for Belief in Uncommon Events

Abstract

The plausibility of uncommon events and miracles based on testimony of such an event has been much discussed. When analyzing the probabilities involved, it has mostly been assumed that the common events can be taken as data in the calculations. However, we usually have only testimonies for the common events. While this difference does not have a significant effect on the inductive part of the inference, it has a large influence on how one should view the reliability of testimonies. In this work, a full Bayesian solution is given for the more realistic case, where one has a large number of testimonies for a common event and one testimony for an uncommon event. A free-running parameter is given for the unreliability of testimony, to be determined from data. It is seen that, in order for there to be a large amount of testimonies for a common event, the testimonies will probably be quite reliable. For this reason, because the testimonies are quite reliable based on the testimonies for the common events, the probability for the uncommon event, given a testimony for it, is also higher. Perhaps surprisingly, in the simple case, the increase in plausibility from testimony for the uncommon events is of the same magnitude as the decrease in plausibility from induction. In summary, one should be more open-minded when considering the plausibility of uncommon events.