This is, somehow, the rational behind Agatha Christie's ``three pieces of evidence''. However it is worth remarking that something is to say there is a rational behind this expression, that can be used as a rough rule of thumb, something else is to take it as a `principle', as it is often supposed in the Italian dictum ``tre indizi fanno una prova''. First, pieces of evidence are usually not `equally strong', in the sense they do not carry the same weight of evidence and sometimes even several pieces of evidence are not enough.25Second, the prior - that is our `0-th evidence - can completely balance the weight of evidence. Finally, we have also to remember that sometimes they are not even completely independent, in which case the product rule is not any longer valid.26
A final remark on the combination of pieces of evidence is still in order. From a mathematical point of view there is no difference between a single piece of evidence yielding a tremendous Bayes factor of ( JL) and ten independent pieces of evidence, each having the more modest Bayes factor of 10 ( JL). However, I have somehow the impression (mainly got from media and from fiction, since I have no direct experience of courts) that the first is considered as the incriminating evidence (the `smoking gun'), while the ten weak pieces of evidence are just taken as some marginal indications, that all together are not as relevant as the single incriminating `proof'. Not only this reasoning is mathematically incorrect, as we have learned, but, if I were called to state my opinion on the two sets of evidence, I had no doubt to consider the ten weak pieces of evidence more incriminating than the single `strong' one, although they seem to be formally equivalent. Where is the point? In all reasonings done until now we have focused on the weight of evidence, assuming each evidence is a true and not a fake one, for instance incorrectly reported, or even fabricated by the investigators. In real cases one has to take into account also this possibility.27As a consequence, if there is any slight doubt on the validity of each piece of evidence, it is rather simple to understand that the single evidence is somewhat weaker than the ten ones all together (Agatha Christie's three pieces of evidence are in qualitative agreement with this remark). For further details see Appendix I.
Giulio D'Agostini 2010-09-30