This is blowing my mind, man.
We have a null hypothesis, which is that there are no elephants in the room. I’m defining elephant to be a standard adult african elephant of typical size and weight. We look in the room, and in so doing we gather evidence. We then determine whether or not this evidence is sufficient to reject our hypothesis. In this case, we’re looking for evidence that there is an elephant. Some examples of such evidence may be seeing the elephant, or hearing or smelling it, or seeing its food or droppings. Other kinds of evidence may exist. If we find this evidence, we determine whether or not it’s probable, given that our hypothesis is that there are no elephants in the room. Assuming we’ve collected some evidence and determined that it’s improbable given our hypothesis, congratulations! We’ve rejected the hypothesis.
If we fail to find any such evidence that there is an elephant in the room, then we cannot reject our null hypothesis, and our next step might be to design another experiment that would cause the hypothesis to be rejected. We may weigh the room, for example. Who knows.
We could also construct the experiment with another null hypothesis; this time, we hypothesize that there is an elephant in the room. We would be looking this time for positive evidence that there’s no elephant in the room in order to disprove our null hypothesis. We look around, and if we are not able to find an elephant, nor are we able to find food or water, then once again we have collected some evidence. Our evidence is that there is no elephant food, or that we cannot hear an elephant, or whatever it is that we’ve discovered and recorded. We consider how likely this would be in the event that our null hypothesis were true and there were an elephant in the room.
Absence of evidence means something very particular. It means you didn’t look, essentially, or that for whatever reason you cannot look in all places. If you have properly defined your experiment, and properly gathered data, then you don’t have an “absence of evidence.” In the elephant example, you could only reasonably have an absence of evidence if you do not go into or otherwise inspect the room. If you did inspect the room, then you’ve gathered evidence and that’s all there is to it.
Absence of evidence is not evidence of absence, it’s just being a poor scientist.