"..and to express the probability of some event having happened in percentage terms is illusory.
....The chances of something happening in the future may be expressed in terms of percentage. ... But you cannot properly say that there is a 25 per cent chance that something has happened... Either it has or it has not. "What is interesting about this is that the judge has used almost the same words that we said (in- Chapter 1 of our book Risk Assessment and Decision Analysis with Bayesian Networks) we had heard from several lawyers. One of the quotes we gave there from an eminent lawyer was:
“Look the guy either did it or he didn’t do it. If he did then he is 100% guilty and if he didn’t then he is 0% guilty; so giving the chances of guilt as a probability somewhere in between makes no sense and has no place in the law”.Of course, as we show in the book (Chapter 1 is freely available for download) you can actually prove that the this kind of assertion is flawed in the sense that it inevitably leads to irrational decision-making.
The key point is that there can be as much uncertainty about an event that has yet to happen (e.g. whether or not your friend Naomi will roll a 6 on a die) as one that has happened (e.g. whether or not Naomi did roll a six on the die). It all depends on what information you know about the event that has happened. If you did not actually see the die rolled in the second case your uncertainty about the outcome is no different than before it was rolled, even though Naomi knows for certain whether or not it was a six (so for her the probability really is either 1 or 0). As you discover information about the event that has happened (for example, if another reliable friend tells you that an even number was rolled) then your uncertainty changes (in this case from 1/6 to 1/3). And that is exactly what is supposed to happen in a court of law where, typically, nobody (other than the defendant) knows whether the defendant committed the crime; in this case it is up to the jury to revise their belief in the probability of guilt as they see evidence during the trial.
David Spiegelhalter points out that the judge is not just 'banning' Bayesian reasoning, but also banning the Sherlock Holmes approach to evidence. But it is even worse, because the judge is essentially banning the entire legal rationale for presenting evidence (which is ultimately about helping the jury to determine the probability that the defendant committed the crime).
p.s. There are other aspects of the case which are troubling, notably the assumption that there were just three possible potential causes of the fire (other as yet unknown/unknowable potential causes would have non-zero prior probabilities). However, the judge got some things right including his line of reasoning about the relative likelihood of two unlikely events (the arcing or the smoking) demonstrated that, if these are exhaustive, then the smoking was the most likely cause.