If you are given a hypothesis "X exists" and you have no evidence for that hypothesis, the rational conclusion is to not believe X exists (which is very different from believing "x does not exist").

How does "not believe" translate into a probability assignment?

Also, the prior is sometimes in favor of existence. There is, at least, a legitimate sense of "evidence" under which I have none for the existence of a person with the initials PQR, but I'm still extremely confident there is such a person.

Also also, precise existential statements must be over domains. The probability I assign to any particular possible finite structure existing in the universe must be at least the probability I assign to the universe being infinite, which is pretty high. Though, of course, I don't have much reason to care whether Zeus exists 3^^^3 light-years away.

I'd like to make an actual independent post on the issue of not believing versus believing not

Please do!