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

absence of evidence is evidence of absence

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    $\begingroup$ Seems reasonable to me. The point of the saying, though, is that $P(B|A^c)$ can still be large (arbitrarily close to $P(B|A)$ in fact). Just because we observe $A^c$ doesn't mean that it is reasonable to believe that $B$ is true; we are just more justified in believing $B$ than we would be if we observed $A$ instead. The crux of the conflict between the math and the saying is that the definition doesn't necessarily capture all the baggage associated with the word evidence. $\endgroup$ – guy Oct 11 '11 at 19:26
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    $\begingroup$ Please shrink the image. $\endgroup$ – zyx Oct 11 '11 at 19:42
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    $\begingroup$ Shrink the image? $\endgroup$ – bodacious Oct 11 '11 at 20:34
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    $\begingroup$ Shrink the graphics in the question, so the huge letters become smaller. $\endgroup$ – zyx Oct 11 '11 at 20:39
  • $\begingroup$ @guy Am I missing something, or did you actually mean "Just because we observe $A$ doesn't mean that it is reasonable to believe that $B$ is true; we are just more justified in believing $B$ than we would be if we observed $A^C$ instead." ? $\endgroup$ – Oren Milman Oct 2 '18 at 17:18
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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).

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  • $\begingroup$ Usually in discussions of probability and evidence, propositions are assumed to be formed in a way that makes the model work. Here $A$ would be something like "I happened upon $\circ$" (where $\circ$ is considered evidence for some state of affairs $\bullet$ being real), in which case not happening upon $\circ$ is simply $\neg\circ$, as intimated. There is an important distinction between a total lack of $\circ$ and merely not happening upon $\circ$, as you say, but under the assumptions of the model the latter is still considered (by hypothesis) to be evidence of $\bullet$, (cont'd) $\endgroup$ – anon Oct 11 '11 at 23:10
  • $\begingroup$ (...) in which case the mathematical argument is completely valid. Your last point is spot-on and I also said it: the inequality deduced from the mathematical argument can easily change if you put additional hypotheses to the right of the conditional $|$, as with our examples, so the inequality itself can be made false in the face of more accurate perspective. (Also, the "necessarily true" qualifier from modal logic I think is too strong to be relevant here, as we're working with a priori "possibly true" statements.) $\endgroup$ – anon Oct 11 '11 at 23:10
  • $\begingroup$ The $\Box$ connective is capable of a variety of interpretations, depending on what it is we want to use our model for. Necessity is the most canonical one, but $\Box$ can also stand for modalities such as knowledge, derivability, eventuality, or almost-certainty -- sometimes with adjustments in axioms and/or formal semantics. The most conspicuous commonality among these interpretations seems to be that $\Box$ acts as a barrier to the rule of excluded middle: $\vdash\Box A \lor \Box\neg A$ is almost never a (meta)theorem. In my remark I'm using $\Box$ mostly to represent "knowledge". $\endgroup$ – hmakholm left over Monica Oct 12 '11 at 15:02
  • $\begingroup$ There was no claim that 'absence of evidence' ⇒ 'absence is probable', which is the paraphrase you use in your analogy. Øyhus is not making anything like the claim you assign to him. Evidence, as defined, is simply something that makes another thing more likely (even if only very slightly so). "I haven't heard or seen rain" is indeed evidence that there is no rain. It may be very weak evidence (I'm blind, deaf or in a basement), or very strong evidence (I'm neither and I'm standing outside), but it is evidence. $\endgroup$ – spookylukey Apr 8 at 12:44
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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:

  • 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.

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  • $\begingroup$ Thanks a lot for your input $\endgroup$ – bodacious Oct 12 '11 at 8:56
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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).

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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).

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  • $\begingroup$ I understand the impetus. This can be tested with hypothetical examples like: if there was life on another planet, but interstellar travel never becomes possible, would that disprove the existence of life on another planet. The answer is clearly no. I don't have much of a mathematical background so was curious as to whether the probability formula was at least sound? $\endgroup$ – bodacious Oct 11 '11 at 20:31
  • $\begingroup$ If the formula is sound, then the discrepancy must lie elsewhere. $\endgroup$ – bodacious Oct 11 '11 at 20:34
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    $\begingroup$ @bodacious: The formula is sound but "evidence" is not the same as "proof". In the life-on-other-planets example, suppose we could only reach one other planet and found no life there. That would be evidence against life on other planets, but only weak evidence, and surely not enough to disprove life on other planets. $\endgroup$ – Charles Oct 11 '11 at 20:54
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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:

  • $B$ 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:

  • 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.
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