From an early age, I've furrowed my brow at the claim that "Absence of evidence is not evidence of absence." Huh? Absence of evidence is not absolute proof of absence, but surely if you don't notice your friend in a room, that's evidence that he's not in the room, right?
[I]n probability theory, absence of evidence is always evidence of absence. If E is a binary event and P(H|E) > P(H), "seeing E increases the probability of H"; then P(H|~E) < P(H), "failure to observe E decreases the probability of H". P(H) is a weighted mix of P(H|E) and P(H|~E), and necessarily lies between the two...
Under the vast majority of real-life circumstances, a cause may not reliably produce signs of itself, but the absence of the cause is even less likely to produce the signs. The absence of an observation may be strong evidence of absence or very weak evidence of absence, depending on how likely the cause is to produce the observation... This is the fallacy of "gaps in the fossil record" - fossils form only rarely; it is futile to trumpet the absence of a weakly permitted observation when many strong positive observations have already been recorded.
Thanks, Eliezer. At last I have an intellectually satisfying response to my friends who continue to believe in Santa Claus. Yes - our failure to find his workshop at the North Pole is indeed evidence that Santa does not exist!