Let us assume the case of psychics and call the hypothesis of a “psychic explanation” H. Bayesian theory tells you to never assign a prior of zero. This is because if P(H) = 0, then no amount of observations can make that hypothesis likely as an explanation.
But it is easy to see that certain observations never make this hypothesis likely as an explanation. For example, suppose a psychic claims to be able to guess a number in my head from 1-10. Suppose he does this five times. Heck, suppose he does this 100,000 times. This may seem, intuitively, to be irrefutable evidence that psychic powers are at work. Or even if you don’t think it is, it would seem to atleast make the psychic explanation more likely.
But does it actually make it more likely? Sure, P (Evidence|No Psychics) may be extremely low here and P(Evidence|Psychics predicting numbers) may be 1, thus resulting in a likelihood ratio favoring a psychic significantly. It seems absurdly improbable to occur by chance. However, this is all useless without a non zero prior for psychics. In other words, without the prior assumption that it is possible for psychics to do their work, this evidence is irrelevant by itself. A zero prior would make this evidence irrelevant.
As per Bayes’ rule, the P (H|E) / P(~H|E) = (P (H) / P (~H)) * ( P(E|H) / P (E| ~H)). Thus, if one does not assign a non zero probability to H, even a million successful predictions become irrelevant. Note that the first term is independent of the second. In other words, a million successful psychic predictions do not increase the prior probability of psychics existing, which is determined independently. It only increases the likelihood of psychics performing something if one already assumes that psychics exist.
Now on what basis can we ever put a non zero prior? As mentioned, we cannot use successful predictions to justify one, since the prior by definition is independent of new evidence. So how else?
It seems that the only way to do this would be to have direct evidence of a psychic. By this, I mean having an actual proposed mechanism for how it is actually being done and being able to empirically observe it. This would justify a non zero prior since one can explain from start to finish how a psychic is actually doing what he does. Only then can one justifiably start looking at psychics as explanations and make it more likely than chance or another explanation.
It seems that without this crucial discovery, psychic explanations shouldn’t even be considered possible, and allow us to justifiably give them a prior of 0 in the meantime. In this case, the zero prior wouldn’t signify that psychics are impossible. Rather, it would from a practical standpoint, signify that it should only be considered possible once direct evidence for psychics existing is observed. This may also reflect a weak point in Bayesian epistemology since the act of observing a mechanism empirically and then updating the prior isn’t part of the epistemology itself. Priors are only updated by what a hypotheses predicts, not by its inherent mechanism of action.
Nevertheless, does this give us a practical case for legitimately giving a theory a prior of 0 in certain contexts?