The question you raise is located not in the pure empyrean region of formal logic but in the much messier and obscure region of informal logic.
Take the phrase: “Absence of evidence is not evidence of absence”
Is this necessarily true?
I don’t think so. It depends on what it is that one is seeking evidence of and in what context. Let’s take a few examples.
If you were to claim that there is an elephant in my dining room I would (carefully) check this claim out. My dining room is finite and relatively small, and there is nowhere an elephant, even a baby elephant, could hide. I look all around and I see no elephant. There is a complete absence of evidence for the presence of an elephant. I would conclude, and I think my conclusion would be reasonable, that the absence of evidence for the presence of an elephant is conclusive evidence of an elephant’s absence.
Now, replace the elephant in our example with a postulated subatomic particle. So far, our finite range of experiments has disclosed no evidence for its existence. Can we conclude, therefore, that it does not exist? I think not. The more we look without finding anything, the more we are inclined to believe it not to exist but it is always possible that it is, as it were, just around the corner.
I would like to see the logical proof to which you refer. I suspect it may be the application of some form of Bayesianism with which, I have to confess, I’m not really au fait. If you do find it, post the link.