Is Absence of Evidence Evidence of Absence?

[Subduction Zone]

Depends on probabilities: imagine opening your wallet and finding no money - is the lack of evidence for money the evidence for the lack of money? But imagine sitting on a train and finding no kids around you, is that evidence for the lack of kids in the city?

Scientists frequently use that phrase to refer to the theoretical impossibility of disproving the inexistence of something, but it doesn't mean that we should treat [close to] zero possibilities as possible.

It sounds as if you are referring to whether absence of evidence is evidence of absence. Though sometimes rejected as a logical fallacy, that is not always the case. The absence of evidence can be evidence of absence. For example a claim from a friend that there was just a huge explosion in city hall in the closest large city to you and a drive past city hall by the person being told that a few minutes later with no sign of damage or any sign of reaction to major damage would be evidence that there was no such major explosion. That is because some events tend to leave evidence of their occurrences. But I would not necessarily put God into that category.