Absence of Evidence

This is a split off a different thread, for this could well be its own but is but a tangent over there.
I assert that, assuming a reasonable expectation of evidence manifestation given the occurrence of an event, an absence of evidence makes a failure of the occurrence more likely than not, and thus more believable than its opposite. Absence of evidence is then, under that condition, evidence of absence, especially when we speak of events that are themselves extraordinary or at any rate rare. The example used last in the topic this is spinning off was the existence of unicorns, so I shall use that example to illustrate my point.


First, let's define a few events. Technically they are sets, but for our purposes it is easier to think of them propositionally:

• Event A: There exist unicorns in a world W.
• Event !A: There exist no unicorns in a world W.
• Event B: There exists evidence of unicorns in W.
• Event !B: There exists no evidence of unicorns in W.

For two events X and Y we define the conditional probability of X given Y as Note, that the union, or AND operation is symmetrical. Thus, if we want to invert the conditional relation, we can easily prove Bayes' theorem:
Now with our events A, B, and their respective negations !A and !B, we can construct several conditional probabilities:
Barring some technicality like the hard problem of scepticism, B almost certainly implies A. So A is (almost) certain, if B is given. I'll use "~" to denote approximations. For simplicity I might work with certainty here, but I prefer a tighter argument. So, seeing as B almost implies A, we can safely say that The converse, as you point out, is not so trivial. Just because we do not find evidence of unicorns doesn't give us a good estimate (yet) as to whether or not W contains any or not. However, unlike the converse, the contrapositive we can speak of safely. If W does not contain any, we are (almost) certain to find no evidence of them: What we want is of course something like P( A | !B ) or P( !A | !B ) = 1 - P( A | !B ). I'll admit here that we will not get there without making further assumptions. So let's see what we need and whether we feel intuitively justified in making them.

Applying Bayes' theorem to the latest postulation we made, we get So the question now is how likely is it that the existence of unicorns would fail to produce evidence, and how likely is each A and !B in isolation. Without making any assumptions, the first equation tells us that the absence of evidence of unicorns is no less likely than the absence of unicorns themselves. In other words, the existence of unicorns is more likely than a recovery of any evidence of their existence, as intuition would lead us to believe. So at least so far it doesn't look like we are doing anything wrong. For the rest, however, we must start assuming things, for we have exhausted what we could do without. Unless otherwise stated, "<" and ">" shall denote "smaller or equal than" and "greater or equal than", respectively.
When we say unicorns exist in W, we of course don't mean that they exist in some abstract philosophical sense with no meaningful impact. It would be much of a waste of our time pondering their existence if it was so clearly indistinguishable from their non-existence. So when we suggest that they exist, we mean to say that if we go out and search for them, we are, after some arbitrary amount of searching, be it measured in time, space uncovered, or what ever other metric we have of the world W, that we would uncover some evidence of them. In other words, under the stipulation, the condition that unicorns exist, we are more likely than not to find a way to confirm that they do, i.e. Now this may be objectionable in principle, but I have made that stipulation earlier in my informal argument and was not criticized on it in particular. If we want, we can rephrase my entire argument to have this as a hinge, a condition. "If we can reasonably expect evidence of an event to manifest given the event", so the argument goes, "then the absence of evidence can be seen as indication of the non-occurrence of the event."
Anyway, if P( B | A ) > 50%, then P( !B | A ) = 1 - P( B | A ) < 50%. What then follows for P( A | !B )? evidently this is not enough, but we are getting there. Now let's consider how likely evidence of unicorns in W is. Well, in the actual world it turns out we have mapped it in its entirety, in part on the surface and in full from outer space. We have had expeditions to almost every piece of land there is to walk and we occupy a majority of what is hospitable to us, too. Assuming that the occurrence of evidence of unicorns was roughly evenly distributed among the forest regions in climates where similar animals like horses would feel comfortable, the likelihood of us finding some evidence of unicorns in what few square feet uncharted territory there remains are frankly rather slim just by the sheer measure of how little remains of the whole. I suggest that P( B ) is not zero but rather quite close to it at this point, let's say P( B ) ~ 0. Then P( !B ) = 1 - P( B ) ~ 1 - 0 = 1.
And that's it! Look at our equation now: Even if the probability of unicorns existing in general is 100%, the probability of them existing and not showing any evidence is still no greater than about half! If the probability of unicorns existing is significantly smaller, then it becomes all the more unlikely that they exist under the condition that there be no evidence of them. The absence of evidence makes absence of the thing more believable, it is literally evidence of absence under these bold but not unfair assumptions. It can get slightly better if we make the assumptions less reasonable. We can say that no, in fact, in the past we managed to look in all the wrong places and that evidence of the unicorns is concentrated in just the few places we haven't looked yet. That way we can try and elevate P( A ) / P( !B ) above 1 and hope that that opens up room for A given !B to be more likely than not. We can also make unicorns more magical and say that they wouldn't leave many footprints or poop or other traces of their existence. We raise that 50% limit that way, but we also make A a more and more specific claim, risking to shrink P( A ) as a result. Or we could leave all notions of pragmatism altogether and say that we are speaking only of some abstract existence that has no consequence to our lives. I concede that this would completely demolish my argument, but at a cost so high that I feel my defeat is the lesser sacrifice in that case.

I find this fascinating, but I am completely at a loss here? What is this system of representation called and where can I learn how to read it?

There are different notations. Particularly my denoting not-A as "!A" here is not especially common.
But anyway, any introductory course on probability theory (or stochastics) will cover the formal aspects of this and quite a bit more. Do feel free to ask for clarification, too, if I have been hopping points too fast here or there. The post turned out lengthy anyway and I was kind of hoping to get to the point moderately soon...

Gisteron,

Thank you this is awesome, I admit this is going to take a few more reads for me over time. I f you have the time and want man I would love to learn more of this. I know I could take a class and may still look into that when I am not taking current graded classes. However, I would just enjoy a nice conversation on this as I continue to learn this. I just wanted to say thank you for teaching me something even if I don't fully get it yet.

Much Love, Respect and Peace,
Kobos

Haha, !A is natural to me from programming

the Tl;dr of it is that there are two disjoint qualities (massive oversimplification): the existence for unicorns ( A ) and evidence for unicorns ( B ).

This means there are four outcomes in a sort of grid

there are unicorns there is evidence for them	there are unicorns there isn't evidence for them
there aren't unicorns there is evidence for them (almost always human error)	there aren't unicorns there isn't evidence for them

the probabilities of each outcome are the product of of the probabilities of each element (because math)

My quick little "absence of evidence isn't evidence of absence" quote is correct (it can happen), but improbable and a bad basis for anything.
We're not a criminal court, so comparing your belief in midichlorians to being accused of murder is lazy

I wouldn't say that A and B are disjoint. If they were, then A would imply !B and B would imply !A.
The probabilities for the intersects of A and B are also only the product of the individual probabilities for A and B if A and B are independent. But if that were so, then that would mean that This contradicts my assertion that that recovering evidence of unicorns heavily implies their existence (barring detection flukes). If A and B were independent, then supposing something about the probability of one would warrant us no inferences at all about the probability of the other.

And yes, there are some programming languages that use the exclamation mark for logical negation. In my experience though it is uncommon in logic and in maths the exclamation mark is normally used for factorials which also show up in probability theory (especially combinatorial analysis) giving all the more reason to use other ways of denoting the negation of a proposition or, as would be here, the "complement" of an event/set.

Minus all the fancy maths that seems to have dazzled everyone into silence (something I consider unfair on your part) I disagree. What you are describing here is a simple probability and not the weight of any sort of actual evidence. Your roaming of the earth in search of unicorns, as a basis for your claim, is a fallacious one unless you can be at all parts of the globe simultaneously. Short of that, the best you can claim is that there were no unicorns at your location at the time you were there. And places like the moon have barely begun to be explored so if a magical unicorn exists it might be there, or even on Mars. Perhaps Elon Musk will finally discover the elusive creatures. Beyond that, all your fancy calculations show is an asymptotic curve that approaches zero to infinity but never actually arrives. What you are trying to claim is that it should eventually cross the zero axis (finally proving unicorns dont exist) and that is just not accurate.

VixensVengeance wrote: What you are trying to claim is that it should eventually cross the zero axis (finally proving unicorns dont exist) and that is just not accurate.

No. That is a lie. I have made the distinction between my usages of "evidence" and "proof" very explicit multiple times over. You have not provided alternative definitions of your own nor presented an argument. I'll be happy to address both if and when you do.

Gisteron wrote:

VixensVengeance wrote: What you are trying to claim is that it should eventually cross the zero axis (finally proving unicorns dont exist) and that is just not accurate.

No. That is a lie. I have made the distinction between my usages of "evidence" and "proof" very explicit multiple times over. You have not provided alternative definitions of your own nor presented an argument. I'll be happy to address both if and when you do.

Lie is a strong word. Do you feel I intended to deceive? At most I would consider it an assertion that you have the right to counter. How about we go with that instead ok? Actually I have not seen this "distinction" you speak of and I dont think anyone else has either. Getting lost in all the math also lost my interest. However by all means please restate your case absent all that and I will be happy to engage further.

Okay, for the most explicit form, here is an excerpt of post #342320 in the old thread:

Gisteron wrote: @VV:
...
This I suppose comes down to what we mean by "evidence" anyway. To make a sloppy, off the top definition, I'm speaking of some circumstance that may contribute to swaying a verdict one way or the other. Evidence is what makes a proposition more believable when opposed to its contrary, than it would otherwise be. When you board a plane, no amount of searching, even including literal slicing the person open and irradiating them with any and all sorts of intrusive radiation can "prove" in a strict logical sense that they have no weaponry on or in their body. No amount of blood tests can determine that someone definitely does not carry a disease that would show up on a blood test. Nevertheless we treat the negative result of even the first or second blood test as (strong) evidence that they do not, and we do not actually slice passengers open in search of firearms or explosives. The absence of evidence is not proof of absence, but it makes the absence (obviously) more believable than it would be if there was not an absence of evidence.

I have applied that preliminary definition of evidence as "that which makes a proposition more believable than it would otherwise be" consistently in this thread, too, as a simple look up at the OP would reveal well before any of the scary intimidating math commences:

Gisteron wrote: ... assuming a reasonable expectation of evidence manifestation given the occurrence of an event, an absence of evidence makes a failure of the occurrence more likely than not, and thus more believable than its opposite. Absence of evidence is then, under that condition, evidence of absence... (emphasis added)

In an even earlier post, I made it very explicit that my usage of "evidence" does not allude to a strict logical implication, that it does not qualify as proof:

Gisteron wrote: If we do perform that search and find no evidence of Frank's cheating, it doesn't of course mean necessarily that he didn't, but it is evidence that he didn't in the sense that ...

Incidentally, that was in post #342257, the one that sparked your initial reply to me about this.

Perhaps I shouldn't have called it a lie. Maybe you were too busy dismissing things for their conclusions (in the other thread) or their icky icky formulas (here) and didn't find it worthwhile to read what you went on to reply to. Or maybe you did read all of those posts and the passages I quote and just forgot. At any rate, I must apologize. Hopefully now that I have brought them all back to your attention/memory, we can carry on discussing which ever substantive weaknesses the argument might have in your opinion.