I know Carl Sagan said the opposite, but he was clearly wrong.



Suppose that a priori X has probability p of being true. We now look for evidence for X of a certain type. Suppose that there is a probablity q that we find this evidence if X is true and probability q' that we find this evidence if X is false. We will assume p<1 (otherwise we wouldn't bother looking for evidence) and that q>q' (otherwise it couldn't be said that the evidence we're looking for is evidence for X).





So we have four possibilities:





X is true and we find evidence for X: probability pq

X is true and we don't find evidence for X: probability p(1-q)

X is false and we find evidence for X: probability (1-p)q'

X is false and we don't find evidence for X: probability (1-p)(1-q')



Under the hypotheses above, the conditional probability that X is true given that we failed to find the evidence is p(1-q)/(p(q'-q)+1-q').Use Bayes' Theorem.Some elementary rearrangement shows this is always less than p given the above hypotheses. It doesn't matter if we are unable to assign an a priori probability, this holds whatever value p has as long as it's less than 1. And if we don't know that q>q' then we shouldn't be in the business of looking for evidence. If the experiment we're doing is any good then q'=0 but as I have shown, the result holds even if we relax this condition.



So clearly failing to find evidence for X should lower our estimate of the probability that X is true.





I wonder what made Sagan say this. I think that maybe he meant to say "absence of evidence is not proof of absence". The theorem shows that under the original hypotheses the conditional probability is never 1, and so while we have evidence of absence, we don't have a proof. But if we can look for enough independent types of evidence it's quite possible for the conditional probability to get close to 1.





Update: The first paragraph was badly written and so I've edited it slightly. The main argument is unchanged. The argument stands or falls regardless of what Sagan actually said, but read the comments below on some context for this quotation. (02 Jan 09)