“my claim is that to reach a desired level of certainty about the burglary being faked, you would need evidence of approximately the same strength required to reach the same level of certainty about murder.”
This assumes that the burglary being faked is the only piece of evidence. If we have three sets of evidence, and each one suggests a 90% chance of guilt, and each is independent of the other, then we have probability (10:1) x (10:1) x (10:1) = (1000:1). No one set of evidence needs to have a (1000:1) probability of guilty in order to reach a final conclusion that the odds are (1000:1). Arguing via modus tollens about a single piece of evidence tells us only that that evidence, in and of itself, is insufficient proof. It tells us nothing about how that evidence may act cumulatively with other pieces of evidence.
(2′) If A is (sufficiently) strong evidence of B, then the prior probability of A can’t be much higher than the prior probability of B.
The logic and math of this post seems very confused. It feels like you are saying “If the sun rises tomorrow, I will kill you. The probability of me being a murderer is 1:10^8, therefor the probability of the sun rising tomorrow cannot be much higher than 1:10^8”
First off, there’s some very crucial evidence you are forgetting in evaluating this case. The key element here is that numerous small bits of evidence are cumulative. This is a very important point, and one which jsteinhardt touched on already.
First, we have a very major piece of evidence: A murder did in fact occur, and the murderer must have been in Perugia at the time they committed this murder. At this point, we have approximately 10^5 possible suspects (Perugia has a population of 166,253), and we know, factually, that one of them is the guilty party. If we had no other evidence, we could reasonably assign a probability of 1:10^5 that each one is guilty. You’ll notice that this is vastly higher than the normal probability of someone being a murderer, because we already have quite a few bits of evidence.
If the burglary was faked with odds of 10^4:1, then we can assume that everyone that had a motive to do so now has a guilt probability of 10^4:10^5, or approximately 1:10. A 10% chance of Amanda Knox being guilty is certainly poor evidence, and I don’t see any reason to favor her over other people who have been demonstrated to have equal motive, but I’m also basing this entirely on this specific post.
The consequences of the burglary being faked does not change based the probability that it occurred, any more than my threat to kill you tomorrow will prevent the sun from rising. If we’re dealing with probability, then there is some factual probability that the burglary was faked, based on it’s own evidence, and this probability is entirely independent of the consequences. Further, this probability, and the probability that (Burglary Faked ⇒ Amanda is Guilty) cannot be 100%, despite your post assuming such. You cannot include impossible numbers and then expect a firm conclusion to arise.
P.S. If your point was simply “The judge is assuming impossible numbers”, then I’d feel you are probably wrong on this point. I’d be happy to elaborate if that is in fact the case.
P.P.S. You can argue that a “higher standard of evidence” for proving that may be required, based on legal and moral principles, but that has nothing at all to do with probabilities.