I once watched a random youtube video which claimed, rather offhandedly, that a lack of evidence was not evidence of a lack. At this point, the physicist in me cringed. I thought about the luminiferous aether, and how the lack of evidence led physicists to conclude that it didn't exist. I thought about string theory, and how it is starting to become somewhat of a joke amongst many precisely because it has yet to produce any solid evidence. It seems to me that in physics at least, a lack of evidence is evidence of a lack. But is this a defensible position to hold, or are physicists just crazy? The track record of physics tells me that there's probably something to this claim.
Enter Bayes' Theorem. Let's use the discrete case for simplicity. After all, we will only need to apply the theorem to two competing hypotheses. In formal notation:
P(A|B) =[ P(B|A)P(A)]/P(B)
Where A represents some hypothesis, and B represents some measurement. Here P(B|A) is the probability of B given A, or the liklihood that we would observe B if Hypothesis A were true. P(A) is a prior probability, which is some non-zero liklihood that we assign Hypothesis A a priori. P(B) is a normalization term, and since it is the same for each hypothesis involved, we can omit it for simplicity. Finally, P(A|B) is the probability that Hypothesis A is true assuming that we did in fact observe B.
So what does all that mean? In a nutshell, given a set of evidence, the probability that some particular hypothesis is correct is proportional to liklihood the hypothesis assigns to the evidence. Applying Bayes' Theorem for each piece of evidence assures us that hypotheses which consistently make accurate predictions have high probability while hypotheses whose predictions are consistently innaccurate have low probability.
Now suppose we have two competing hypotheses. One hypothesis, which we'll call A, is that a secret society of Neo-Nazis has infiltrated American society and is waiting for just the right moment to strike. The other hypothesis, which we'll call A*, is that no such secret society exists.
Now, what is the probability of America not getting attacked by a secret society of Neo-Nazis, assuming this society does not exist. That's obvious. If there is no society, there can be no attack, and so the probability of no attack is 100%, or P(B|A) = 1, where B is "no attack."
And what is the probability of America not getting attacked by a secret society of Neo-Nazis, assuming this society exists and intends to attack us. Okay, sure, they might want to lay low for a while and remain secretive, but eventually they want to attack us. So there is some chance, however small, that we will get attacked, assuming this secret society exists. So the probability of not getting attacked is strictly less than one, or P(B|A*} < 1. And this is true no matter how secretive the society is.
So lets start with a completely unbiased stand. At the outset, we'll assign a 50/50 chance to the society's existence. Then, we observe that over the past five years, we have not been attacked by Neo-Nazis. We plug all this into Bayes' Theorem and we find:
P(A|B) = [1*50%]/P(B) while P(A*|B) < (1*50%)/P(B)
So given this lack of evidence for the existence of the Neo-Nazis, we conclude that hypothesis A is in fact more likely than hypothesis A*. In other words, no matter how secretive or elusive this society is supposed to be, the lack of evidence is still evidence of a lack.
And what works for secretive societies of Neo-Nazis works for gods too (a fitting analogy?). If you believe in a god that does anything, a deity that isn't so removed from the world as to be basically nonexistent, then there should be some non-zero chance of seeing some effects of this deity. So if we look and look and keep not finding these effects - we keep not seing amputees healed by prayer, we keep not seeing believers being reliably more moral than non-believers, we keep not seeing any intervetions on behalf of those in trouble who pray for help - what we find is an enormous heap of evidence against this god hypothesis.