Absence of evidence
Absence of evidence, or the failure to observe evidence that favors a hypothesis, is evidence against that hypothesis. This is because we are significantly more likely not to see evidence for a hypothesis when it is false than not to see it when it's true — some assertions demand that the universe be screaming with supporting evidence, so when that evidence is not actually observed, it counts against it. The idea is often, but fallaciously, summarized as 'absence of evidence is evidence of absence' (or non-fallaciously as 'absence of evidence is not evidence of absence').
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Visual explanation
Blue dots: Universes where we lack evidence (E)
Orange circle: A sub-set of universes where H is true and supported by E. 56% of these exhibit E.
Green circle: All universes where H isn't true, but we may observe E anyway at about 5% chance.
The conclusion is that not observing E is evidence that H is most certainly not true, i.e., we're in the green circle, not the orange circle.
The diagram on the right demonstrates why the absence of evidence confirming a hypothesis is a reason to lower your confidence in that hypothesis. In the diagram, the green circle represents the space of possibilities — think of it as a set of possible universes, and we don't know which one we're in. The orange circle represents the group of possibilities in which a hypothesis — let's call it H — is true. A red dot represents a universe in which we observe some piece of evidence — let's call it E. A blue dot represents a universe in which we do not observe this piece of evidence.
Looking at the diagram, it is clear that observing E is evidence in favor of H. This is because there are proportionally more instances of observing E inside of the orange circle, which represents H, than there are outside of it. In other words, if you see E then you know you're in one of the red circle universes, and because there are more red circles in H than outside of it, it's more likely that H is true. Conversely, not observing E means that you are in one of the blue circle universes, so it is more likely that you are outside of H. This means that failing to observe E is evidence against H, because it decreases the probability of H being true.
Technical explanation
"Absence of evidence is evidence of absence" is not just a rule of thumb — it can be formulated as a provable theorem of probability theory. If there is some hypothesis H and some event E such that P(E|H) > P(E), we know that P(H|E) > P(H), or "E is evidence in favor of H." If this is the case, it is also true that P(~E|H) < P(~E), so we can conclude that P(H|~E) < P(H), or "not observing E implies lower probability of H." A quick proof is as follows:
By the formula of total probability:
Hence, by the definition of conditional probability:
This simply means that P(H) is a weighted average of P(H|E) and P(H|~E) and is therefore somewhere between these values. Therefore if P(H|E) > P(H), i.e. if E is evidence for H, then P(H|~E) < P(H), i.e. ~E is evidence against H.
Examples
You hear some rustling noises in your backyard, and you want to figure out if the noises were caused by your neighbor's dog or by some other intruder. Fortunately, you know what your neighbor's dog is like — he's rambunctious and he barks constantly. So you listen closely, and after 20 minutes you don't hear any barking. What should you conclude? If you had heard a barking noise, that would have been evidence for the neighbor's dog being there. Thus, the absence of barking is at least weak evidence that the noises were caused by something other than the dog. Even though your neighbor's dog might still be the most probable cause of the noise, the absence of barking is evidence against that hypothesis.
Take another even more obvious example. Someone maintains that they keep an elephant in their garage. A perfectly normal visible large grey elephant. You go to look and you cannot see any elephant. You enter the garage, which is quite small, and look around it. There is no elephant smell or any elephant droppings. In fact there is a total lack of evidence of any sort which would suggest there is an elephant in the garage. You would quite reasonably assume that the absence of elephantine evidence indicated the absence of the elephant.
Caveats of probability
There are a few caveats to take into account to refine what a lack of supporting evidence says about a hypothesis. Absence of evidence is not necessarily strong evidence that outright disproves the hypothesis in the way that an observation that contradicts the hypothesis would be. For example, in Russell's Teapot, searching the solar system and not finding the teapot doesn't outright disprove that it isn't there as one may have simply not looked in the right place at the right time, and this problem is the basis behind falsifiability. Similarly, particle colliders failing to detect the Higgs boson in a single experiment doesn't mean the particle certainly doesn't exist, as formation of these particles is a rare and high energy process and we would expect a significant number of null results.
As such, absence of evidence acting against a hypothesis is only a probabilistic approach and works best in a full Bayesian-style framework, which also takes into account other probabilities and other evidence. In the example invoking Russell's teapot, the odds of a celestial teapot existing are remarkably low so absence of evidence can be used to dismiss the teapot's existence with a good degree of certainty. When other factors make the hypothesis more plausible, then the lack of a specific piece of evidence is significantly less able to dismiss a hypothesis. For example, it's often asserted by creationists that a lack of certain transitional fossils disproves evolution, but these individual specimens may be missing because fossilization is a rare process and not every single specimen has been discovered and cataloged. The probability of evolution being true based on other evidence is high enough that a lack of a specific transitional fossil can't call it into question.
Because there is always this faint possibility that evidence hasn't been observed yet, a common maxim is that "absence of evidence is not evidence of absence" — and is often used by people to hang on to their beliefs even when faced with a lack of evidence for them. However, this is technically an incorrect maxim; if evidence is lacking when we expect it to be abundant, then it very much allows us to dismiss a hypothesis, and absence of evidence is evidence of absence.
Side note
As an important side note, there are plenty of cases of affirmative Evidence of Absence that have nothing to do with any Absence of Evidence: Consider, for example, two chemicals that react violently with each other. The presence of a large deposit of one of these chemicals would be affirmative evidence of the absence of the other.
One prominent case: the currently cited (as of 2019) best evidence of the existence of dark matter (as opposed to various so called "MONDs", which usually attempt to explain the same phenomena without any "dark matter") is a set of galaxies that appear to have none. (The details are too complicated to go into here, but the short version is that they appear to spin exactly as if their mass was equal to the sum of their stars, while all other galaxies have apparent masses much greater than what can be summed from their observable stars.)