Absence of evidence

Equivocation much?

Ever since Donald Rumsfeld uttered the antimetabole in a righteous press conference regarding Iraq not possessing any WMDs

"There's another way to phrase that and that is that the absence of evidence is not the evidence of absence. It is basically saying the same thing in a different way. Simply because you do not have evidence that something does exist does not mean that you have evidence that it doesn't exist."

an abbreviated form of it is often cited as if it is an established truism:

Absence of evidence is not evidence of absence.

The mind boggles.

But let's consider that proposition for just a second, and ask, "What then, is evidence of absence?"  How can we ever conclude any of these:

Rats in the attic.
Do you have rats in your attic? If so, I presume you'll exterminate them. If you are successful, you'll have no rats in your attic. Rats will eventually be absent from your attic. But how do you establish their absence and stop with your extermination procedures?

Canada has WMDs and is planning to invade Michigan and use them.
Should we do something, like we did in Iraq? Why not? If Canada is planning to invade Michigan, it is surely worse than the Iraqis just possessing WMDs. How can we determine that Canada does not have WMDs and that they are not planning an invasion?

A Vatican priest exorcised demons out of a boy's body.
I admit that it is a curious assumption that there were demons in the boy to begin with, but how did they determine that the demons were gone?

In each of these three cases, if we are to claim we have evidence of absence and that such is not absence of evidence, what is it?

Or maybe, absence of evidence is evidence of absence after all. Of course, it is not proof, the stricter term often thrown about carelessly colloquially as if what one really meant to say was absence of evidence is not proof of absence. Which is fine. But also quite useless. Unless you provide evidence that such was necessary in each of the three cases above.