The common saying "Absence of evidence is not evidence of absence" is not true.
If you should be seeing evidence of a phenomenon and you aren't, then that is evidence that the phenomenon doesn't exist. Consider the Michelson-Morley Experiment. It was designed to measure the speed of the Earth through the luminiferous aether. And it found... absolutely nothing. That, and other experiments which failed to detect the aether, overthrew the theory. The absence of evidence was the evidence of absence.
Even in cases where you wouldn't expect to see evidence, the absence of it is still weak evidence of absence. I say this based on Bayesian reasoning. If a phenomenon exists, but you wouldn't expect to see evidence of it given the circumstances, presumably, there's still a non-zero (though small) probability of seeing evidence of it. Whereas, if the phenomenon doesn't exist there is even less probability of seeing evidence for the phenomenon. This means that not seeing the phenomenon does shift the probability of the phenomenon actually existing down, by however small an amount. Of course, depending on the specifics, it could be a very, very weak evidence.
Usually, this sentiment comes up in reference to the existence of god. Just because we don't see evidence of god doesn't mean he doesn't exist. But it does make it less likely. The question then is, how much less likely? If god did exist, what would we expect to see different than if he didn't exist?