Okay, I am going to try to plumb this Bayesian inference thing from the POV of an electrical engineer. In communications systems we use Bayesian inference regularly to postulate what symbol was originally transmitted when some particular signal was received. I think, when Bayesian inference is used in science, it is to consider the likelihood that some hypothesis is true when some other evidence is observed. The communications problem of the electrical engineer is simply a specific example of the more general inference of hypothesis, given evidence, used in science.
Now, if we call a specific hypothesis, H, and some set of evidence (that may or may not support the hypothesis), E, then Bayes rule (along with some other quantitative axioms of probability) says this:
Now, I would like you to focus on the third line of the conditional probability, P(H|E). The quantitative value depends completely on the second term of the denominator:
Now, if that term is equal to 1, then the entire dependent probability is P(H|E) = ½ and the odds (given evidence E) are equal and it's just as likely that fairies exist as not. If that term gets "astronomically large" in value, then it's astronomically unlikely that the hypothesis, H is true. But if that term gets extremely small, very close to zero, then the likelihood that H is true, given this evidence E.
So this is how a Bayesian reasoner would frame the debate. Now, of course, it depends on what the evidence E is.
The OP posits that there is no evidence that fairies exist. Well, I would like to, just for the sake of argument, say that perhaps that there are books that say something about fairies, that this be treated as "evidence". Now, it seems to me that the fact that these books that say that fairies exist, that this does not help the case for the existence of fairies.
Now, with the fairy hypothesis, let's say that the only evidence to support fairies is that there are books that talk about fairies. Now we have books that talk about George Washington. I have never seen George Washington nor have seen direct evidence of the existence of George Washington other than what I see in history books and in modern political discussion of American history. It is my judgement that P(E|¬H) is much much smaller than P(E|H) so, even if I start out with the initial belief that P(H)>0, but is not ≈0 (that the a priori likelihood that George Washington existed is not astronomically small) then all of that contributes to the value of that term above being very close to zero, so I judge that, given the evidence of George Washington that I read in the literature (and in all other references to the first president), the hypothesis that George Washington is an actual historical person is quite likely true. P(H|E)≈1 . It's because if Washington did not exist, then the likelihood of seeing this evidence is much smaller than the likelihood of seeing it if Washington did actually exist.
But with fairies, the evidence referring to and describing fairies in books would exist almost as likely whether fairies actually existed or not. In that case P(E|¬H) is about the same as P(E|H), so that ratio is about 1. So for me to conclude that fairies exist given the evidence in the literature, I would have to start with a reasonably strong a priori likelihood that fairies exist P(H)≈1. But if I start out with P(H)≈0, then the "evidence" in the literature does not help.
Now, however, suppose there existed some magical pixie dust, that had supernatural properties, as evidence E? That would change things. That is because P(E|¬H) << P(E|H) and that ratio becomes very close to zero. So then, for me to conclude that fairies don't exist, I would have to start with the a priori likelihood of fairies existing P(H) being astronomically small to begin with. So, in other words, if you are staring at evidence of pixie dust with supernatural properties, for you to conclude that fairies don't exist, in the face of this evidence, would require a strong a priori belief that fairies don't exist to start with. I would consider that to be a prejudice.
Now, in my opinion, a better hypothetical scenario to consider is more like this: Suppose you are seated at a poker table for the very first time in your life and you are dealt, for your very first hand, a Royal Flush in hearts. What are you gonna think? That you're a superb poker player? That you're extremely lucky? Or, simply based on the probabilities, that maybe someone is stacking the deck (and maybe whoever did that really likes you)?
The likelihood of a Royal Flush of a specific suit is 1 outa 2598980. That is P(E|¬H) where E is the fact that you're dealt this Royal Flush and H is the hypothesis that someone is stacking the deck. If those probabilities are not sufficiently astronomically low, consider the hypothetical that one person wins the Lotto six or eight times in a row. It's not impossible, but if it were to happen, based solely on the probabilities, all of us would reasonably suspect that someone is nefariously fixing the game.
I believe that the original intent about the original post is not so much about fairies, but is more about the existence of God. If that is the case, might we discuss this Bayesian reasoning in light of the intended question beneath the surface?