Absence of evidence is not evidence of absence.
In many cases that is good advice. But not all. Say you have a three-foot by three-foot cardboard box, and someone tells you that there is a red ball inside that measures three feet. But you open the box and see nothing inside. There is clear absence of evidence that a ball is inside. You can do all sorts of tests and experiments to detect the presence of the ball, in case it is invisible. But no evidence of its presence can be obtained. Does that mean that this is evidence that the ball is absent? I think nearly anyone evaluating the situation would agree so. So a better more nuanced way of stating the proposition is this: "Absence of evidence is not evidence of absence unless there is reason to specifically expect the presence of evidence."
Look at the Trevor/Linda debacle. There is reason to specifically expect that if Linda exists, there would be evidence of that existence. If she was married to Trevor, there should be a marriage license. She should have a birth certificate. If these don't exist, then one is left with two possibilities: (1) Linda doesn't exist, or (2) There must be some unusual circumstance that would make all records of Linda's existence vanish. If there is no viable explanation for (2), then Occam's Razor would naturally lead one to conclude that absence of evidence is evidence of absence.
While the advice may be true in formal black-and-white deductive logic, we more commonly evaluate evidence probabilistically through induction. So one may conclude from one's observations that zebras have black and white stripes. But there is a leap in logic there because one has not examined every single zebra in existence and which has ever lived to see if there are exceptions to this. But the absence of blue-and-red striped zebras and the absence of no-striped zebras is enough to convince a person that the statement that all adult zebras have black and white stripes is probably true. It is not a formal proof, but a reasonable generalization. So if one makes a proposition that, if true, should be supported by evidence with reasonable probability, and if that evidence does not exist, then it is reasonable to infer that the proposition is false. Hence, we are able to judge the statement that "Zebras have pink and purple polka dots" is false without having to physically examine every single zebra in the world to be sure.
On account of this, it is useful to have a burden of proof. If a generalization is not known to have any exceptions, the burden lies with the person wanting to falsify the generalization. If a person believes that zebras can have pink and purple polka dots, it is not reasonable for her to demand a skeptic to prove that every single zebra in the world doesn't have pink and purple polka dots. The burden of proof is on the believer to submit evidence that zebras can in fact have colored polka dots. Similarly, the burden of proof is on the believer to substantiate the claim that the Loch Ness monster exists, or Santa Claus, or the god Thor. Is the burden of proof on the person to prove that the Loch Ness monster is a myth, or on the person to demonstrate that this creature exists in the first place?
As far as God is concerned, faith is not a matter of empirical facts and evidence. It explicitly is belief irrespective of evidence, "faith is being sure of what we hope for and certain of what we do not see" (Hebrews 11:1). It is a conclusion that is orthogonal to evidence, it furnishes its own proof for the believer (this also means imho that matters of faith are not something to be proved through empirical evidence). That is a rather different philosophical approach than reaching an unprovable conclusion (like the belief that no pink and purple polka dotted zebras exist) through induction; the two approaches are not the same. Or some reinforce their faith with their own subjectivity, i.e. (from their own POV) how God touched their life. But a person who appeals to their own subjective experience (and which is more than sufficient for him/herself) alone has access to it; others cannot evaluate it and accord it objective value. Many atheists expressing strong atheism may make an inductive leap, but that is no different than concluding that the Santa Claus does not exist. Weak atheists (= agnostics in common parlance) are open to the possibility of God's existence but recognize (in probabilistic terms) that its likelihood is rather low, such that non-existence is judged as more probable than existence. Atheism here is literally a-theism, i.e. no theistic belief. No specific belief in a God as opposed to a specific belief about God not existing. If God dramatically intervenes in the world's affairs in an undeniable way, then a specific belief (in favor of God's existence) would be formed under those circumstances. Just like in the case of Santa Claus. You developed your (strong) conclusion that Santa doesn't exist on the basis of absence of evidence. But if suddenly objective evidence of Santa's existence presented itself on this Christmas Eve (i.e. observable and testable to everyone, NOT subjective evidence that someone else could dismiss as a hallucination), then you obviously would have to revisit your initial premise that Santa doesn't exist. Similarly if someone discovers a real live pink and purple polka dotted zebra. But until those things happen, the conclusion that is most probable in view of the evidence is the one that is usually preferred. Now different people will inevitably evaluate the evidence in different ways, but regardless of this, the burden of proof remains on the person asserting than an (objectively) unobserved entity exists. Just as the burden of proof rests on the Watchtower Bible & Tract Society that it an instrument of an unseen "faithful and discreet slave".
BTW, I would classify myself as a weak atheist....I am not willing to dismiss others' subjective experiences (while at the same time not treating them as useable, objective evidence for myself), indeed I respect others' faith even if I reach other conclusions, and I would be glad to revise my evaluation of things if the evidentiary situation changes, but at present I have no specific belief in the existence or non-existence of God...though I view the latter as more probable than the former. At the same time, I am very interested in beliefs as a social phenomenon and how they reflect changes in society and intellectual development. There is a lot that can be learned culturally if we try to understand what others believe and why they believe.
