Evidence of Absence = Absence of Evidence? Any clever-cloggs out there who can explain the formula below in more simple English please? - Do you agree with the formula?

 A: An event A is called evidence for an event B if the latter is more likely in light of the former than in absence of the former. For example, upon discovery of my dead corpse one should consider the event "anon was murdered" more likely than if I were instead discovered breathing and happily typing away on my keyboard or, alternatively, I was not found at all (in which case I could still be quite alive but perhaps in hiding). Put this way, the saying does seem fairly tautological, but there are two common and very serious issues when actually trying to put it into play in the real world:


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*How much more probable "absence of event B" is in view of the absence of event A might be vanishingly small, and essentially moot in comparison to more significant information also available: you may not have discovered my corpse, which is technically evidence for my not being murdered by the mathematical definition, but if you also receive a video through the mail depicting my gruesome murder by the Joker, the lack of corpse is rendered meaningless.

*Perhaps the event A doesn't actually meet the mathematical criteria of "evidence," but every-one's gut feeling is that it does: it may be perfectly consistent with B and easier to process along-side belief in B than belief in not-B. For example: dramatic stories that shine one's own country in a negative light might be considered less probable by the citizenry if there is a total absence of news reporting on the aforementioned stories (shouldn't we expect those stories to be disseminated in some fashion?) but this is fallacious reasoning because maybe there is a systemic reason for their being no reporting; perhaps news is being suppressed by national interests and so the assumption that reports are serious evidence is wrong.


Ultimately it boils down to information and perspective. Maybe we don't have the information necessary to make the most confident of judgments, in which case any calculation of probability is volatile in the sense that it could change quickly in light of Bayesian update through new (and not necessarily all that surprising) information. And maybe a lack of perspective clouds one's judgment, allowing one to think a hypothetical fact would be meaningful evidence when in fact the story could be rather more complicated, and basic observations superficially fulfill false preconceptions.
Addendum: As Henning points out, there is also an important difference between an actual absence of a particular form of evidence and merely not happening upon said evidence - if our probabilistic model refers to the former, then it won't be realistic because rarely can we check that no evidence is to be found anywhere (we aren't omniscient), whereas if it refers to the latter it is considerably weaker because we can't take $\neg A$ as an established fact but rather only as an educated guess.
Moreover, my bullet points and emphasis on perspective and information can be given a mathematical backing: even though the argument is sound and conclusion follows, it's also the case that more information can change the inequality the other way around:
$$P(\;\neg B\;|\;\neg\text{ evidence } A)>P(\;\neg B\;|\text{ evidence }A),\text{ but}$$
$$P(\;\neg B\;|\;\neg\text{ evidence } A,\text{ info }X)<P(\;\neg B\;|\text{ evidence }A,\text{ info }X).$$
We don't operate in a vacuum, so the possibility of additional information throwing off our calculations is real. More pointedly, one thing it does not establish is that $\neg B$ is more likely than $B$: in other words, even if $A$ is evidence for $B$ and its 'absence' is evidence for $\neg B$, it may very well be swamped by other facts so that $B$ is still likely to be true even if we're lacking the evidence $A$ for it, as in the example where others lack my corpse but it's still vastly more likely (in view of the video) that I was murdered (technically, I might have staged and edited it though..). Symbolically,
$$A\text{ evidence for }B\not\Rightarrow P(\neg B|\neg A)>P(B|\neg A).$$
All in all, taking this saying seriously in the real world is practically guaranteed to be hazardous without an accurate perspective and a much more in-depth understanding of context.
A: As far as I can make out, it's just mincing words. Logical negation is not a good representation of what "absence" means. I'll assume that it is standard to describe the observation of $A$ as "evidence"; given the conditions of the poster this seems to be at least somewhat reasonable. However, then "absence of evidence" does not mean observing $\neg A$. It simply means that we're not observing $A$ -- which could be because both $A$ and $\neg A$ are events that cannot be, or have not been, observed directly.
"Absence of evidence" can also be taken to describe the situation that we can identify no random observable that is even (provably) dependent of $B$. This too is very different from observing such a variable and find that it doesn't support $B$.
In essence, the author is committing the very fallacy he's denying. He's assuming that the phrase "absence of evidence" ought to be modeled mathematically as a construction that actually models presence of evidence to the contrary.
Or, in modal logic terms, $\neg \Box A$ is not the same as $\Box \neg A$.
(Since no post is complete without a bad analogy: Consider A="it is raining" and B="the train will be crammed full". A is indeed "evidence" of B in the poster's sense, because when the weather is bad more people will be likely to take the train instead of cycling to work. Then "evidence" would be looking out the window and seeing it pour down, in which case could validly conclude that the risk of a full train is higher than usual. "Absence of evidence" would be being in a windowless basement and not knowing how the whether is. You can't reason from "I'm in a basement" to "the train is probably not full". Øyhus seems to be claiming that "absence of evidence" implies being certain that the weather is dry).
A: If you would expect to notice something if it was there (like an elephant standing in your room), and you don't notice it, then that means it's probably not there (there's no elephant in your room).
A: If you read a bit on Oyhus' homepage, you can see that the impetus for this proof is at least partly as a response to statements like: "Even if there is no evidence for the existence of God, it does not mean God does not exist."
Strictly speaking, the proof isn't a response to this statement at all. The trick lies in a bit of the persuader's trade in blurring the distinction between "evidence for non-existence" and "proof of non-existence".
What's really shown in the proof is that lack of evidence for a claim makes it more likely the claim is false (i.e. evidence for non-existence). This is a far cry from being certain the claim is false (i.e. proof of non-existence).
A: Øyhus wrote this proof because he noticed that:

People quite often state wrongly that "Absence of evidence is NOT evidence of absence."

To correct this misconception, he showed in this proof that the opposite is true for the following case:


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*$B$ is a binary statement that we care about, i.e. we wish to know whether it is true.
E.g. we define $B=\text{"Dragons exist"}$.

*$A$ is a binary statement about new information that we encounter. (It is tempting to use the phrase "new evidence" here, but "evidence" has a special meaning, so I use "new information" instead.)
E.g. some "Game of Thrones" fans conducted an expensive search to find dragon fossils over the past year. We define $A=\text{"The fans found fossils"}$.


Øyhus defines "absence of evidence" as $\neg A$, i.e. the statement about the new information is false.In our case, "absence of evidence" would mean that the fans didn't find fossils.
(I agree with Henning - this interpretation of the phrase "absence of evidence" is not intuitive, and it is even more confusing here, as "is evidence of" is also defined to have a special meaning.)
Øyhus defines "absence" as $\neg B$, i.e. the statement that we care about is false.
In our case, "absence" would mean that dragons don't exist.
I think that getting familiar with Bayesian inference would help understand Øyhus' definition of "$A$ is evidence of $B$", so here is a short and very partial introduction, with regard to our case:


*

*$P(B)$ is our estimation of the probability that the statement we care about is true, before taking into account the new information. As this is our estimation before considering the new information, $P(B)$ is called "prior".
E.g. $P(B)$ is our estimation of the probability that dragons exist, without taking into account whether the fans found fossils or not.

*$P(B|A)$ is our estimation of the probability that the statement we care about is true, assuming that the statement about the new information is true. As this is our estimation after considering the new information, $P(B|A)$ is called "posterior".
E.g. $P(B|A)$ is our estimation of the probability that dragons exist, assuming that the fans found fossils.
Similarly, $P(B|\neg A)$ is also a posterior, but it is our estimation of the probability that dragons exist, assuming that the fans didn't find fossils.

*Note that after considering the new information, we should update our estimation of the probability that dragons exist to either $P(B|A)$ or $P(B|\neg A)$ - according to whether the fans found fossils.
I.e. the relevant posterior would be our new prior next time that we encounter other new information.


So Øyhus defines "$A$ is evidence of $B$" as the case in which the posterior that assumes that $A$ is true is higher than the one that assumes that $A$ is false.In other words, "$A$ is evidence of $B$" means that we will end up with a higher estimation of the probability of $B$ if $A$ is true.
E.g. $A$ is evidence of $B$ means that $\text{"The fans found fossils"}$ is evidence of $\text{"Dragons exist"}$, which means that our estimation of the probability that dragons exist will be higher if it turns out that the fans found fossils, than if it turns out that they didn't.

Finally, we can turn to the proof itself.
Øyhus didn't mention that, but the proof takes as a given that $A$ is evidence of $B$, i.e. $P\left(B|A\right)>P\left(B|\neg A\right)$.
(At least in our example, $A$ is evidence of $B$ indeed seems quite obvious.)
Øyhus concludes the proof by showing that:


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*By the complement rule: $P\left(B|A\right)>P\left(B|\neg A\right)\leftrightarrow1-P\left(\neg B|A\right)>1-P\left(\neg B|\neg A\right)$

*By simple algebra: $1-P\left(\neg B|A\right)>1-P\left(\neg B|\neg A\right)\leftrightarrow P\left(\neg B|A\right)<P\left(\neg B|\neg A\right)$

*By the fact that $A=\neg\neg A$, it holds that $P\left(\neg B|A\right)<P\left(\neg B|\neg A\right)\leftrightarrow P\left(\neg B|\neg A\right)>P\left(\neg B|\neg\neg A\right)$

*By Øyhus' definition of "evidence", $P\left(\neg B|\neg A\right)>P\left(\neg B|\neg\neg A\right)$ iff $\neg A$ is evidence of $\neg B$.

*By Øyhus' definitions of "absence of evidence" and "absence", $\neg A$ is evidence of $\neg B$ iff absence of evidence is evidence of absence.

