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August 11th, 2015, 11:09 AM   #1
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Metaphysical Probability Formula

I’m looking for a probability formula that is based more on theoretical concepts rather than defined numbers, and I’m not quite sure it exists. I’ve looked through the link below (there are lots of different probability formulas listed at the bottom of the webpage) but none of them seem to be quite what I need.

Link: Probability Formula

As an example of what I’m needing this probability formula to be for, consider the following two claims:
1) I went into the woods and saw a squirrel.
2) I went into the woods and saw bigfoot.
On their own, each claim could be true or false, but in relation to background evidence both claims are not equally probable.

For the first claim, the general existence of squirrels is well documented and supported by indisputable, direct evidence, which means that the first claim has a high probability of being true at face value. That individual claim could be further investigated and it could be found that I was lying about seeing a squirrel in the woods, but nonetheless it’s reasonable to believe the claim upon hearing it based on the large amount of background evidence which supports the existence of squirrels.

The second claim, however, is only supported by disputable, indirect and at times discredited evidence. There have been many claimed sightings of bigfoot, but none have been confirmed. Because of this, even if I truly believed that I saw bigfoot, that claim on its own is not likely to be true given the large amount of background evidence that does not support the general existence of bigfoot. Even if in objective reality I did actually see bigfoot, I shouldn’t expect anyone else to believe that claim until I had strong evidence that overrides the current background evidence that makes it improbable that any individual claim of seeing bigfoot is true.

So, given that we don’t know exact numbers for claimed sightings of squirrels or bigfoot to use as a denominator, it makes it kind of difficult to apply a defined formula to those claims. Could something be set up to where maybe 1 = “Supported by direct evidence” and 2 = “Not supported by direct evidence” and those numbers are used as denominators? The outcomes may or may not line up with traditional statistics with, for example, 0.5 being a 50% chance of the claim being true, but that’s ok so long as the formula could be applied consistently to any claim. So if anyone knows of an already existing formula for this problem or has ideas on how to create one, please feel free to respond!
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August 11th, 2015, 11:47 AM   #2
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Bayesian statistics could be useful. See, for example, this introduction:
Kevin Boone's Web site

But for cases like Bigfoot where no instances have been convincingly reported you'll need something like Laplace's 'sunrise problem' approach. For example, you might say that there have been (say) a million attempts to observe Bigfoot, with no successes; adding one fictitious observation and one fictitious non-observation, this suggests that the chance of observing Bigfoot is at most 1/1000002.

It's dicey, but that sort of thing might be the best you can manage on that kind of data.
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August 11th, 2015, 12:30 PM   #3
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I think there is nothing wrong or lacking with conventional statistics and probability theory.

The simple fact is that we can say for certain that there is a probabilty value but that we do not know what it is in the case of the bigfoot.
We just know what it is not.

That is because we cannot rule a bigfoot sighting or the existence of bigfeet.
Since we count the possibility of such an animal as possible, but never observed, the sighting probability cannot be zero.

There is nothing wrong with the phrase 'insufficient data'.
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August 11th, 2015, 09:49 PM   #4
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CRGreathouse, thanks for the link to the Bayesian statistics stuff and suggesting the Sunrise Problem. I’ll research those and see if they will work for abstract claims, but it could end up being that metaphysical beliefs can’t be adequately expressed as mathematical functions, maybe similar to how a poem can’t be adequately expressed as musical notes. I’m hoping there’s a connection between the two though!

And I understand your example of a million-to-one odds of a particular bigfoot claim being true, but the tricky thing is creating a formula/rule that allows for a million unconfirmed sightings to be overruled by a single instance that’s supported by direct evidence, say if someone were to kill or capture a bigfoot, for example.


And studiot, I agree that there’s nothing wrong with having insufficient data, but the issue becomes problematic when people disagree on how sufficient the data is. Many people have evaluated the same data regarding bigfoot's existence, and some have concluded that it supports bigfoot's existence while most have concluded that it does not. It would be helpful to have some sort of objective measure to show who is more likely to be right or wrong when it comes to beliefs like this.
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August 12th, 2015, 02:07 AM   #5
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Quote:
It would be helpful to have some sort of objective measure to show who is more likely to be right or wrong when it comes to beliefs like this.
The objective measure supports the conclusion of insufficient data.

Wishful thinking does not make it otherwise.
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August 12th, 2015, 06:54 AM   #6
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Quote:
Originally Posted by The Power View Post
And I understand your example of a million-to-one odds of a particular bigfoot claim being true, but the tricky thing is creating a formula/rule that allows for a million unconfirmed sightings to be overruled by a single instance that’s supported by direct evidence, say if someone were to kill or capture a bigfoot, for example.
This is the part where Bayesian statistics are useful. Let's say that your prior is that the chance that Bigfoot exists is 1/1000002 based on the crude 'sunrise problem' statistic above (and my made-up claim about the number of observations). Now a Bigfoot carcass surfaces. To apply Bayes' theorem, you need to find the probability that a Bigfoot carcass is found -- let's call this P(B) -- and the probability that a Bigfoot carcass would be found, given that Bigfoot exists -- let's call it P(B|A). Then the updated probability that Bigfoot exists, given that you've found a carcass, is 1/1000002 * P(B|A) / P(B). Since you think it was unlikely that Bigfoot existed, you should also think that P(B) is small, in particular at most 1/1000002. Let's say 1/1100000. But if Bigfoot really does exist, the probability of finding one seems decent, let's say 90%. So those numbers would suggest that our updated probability that Bigfoot exists jumps to about 99%. (Of course, you can tweak this with your own numbers.)
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August 12th, 2015, 11:00 AM   #7
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Ok, I think I've got it. Except what I have is not based on number of sightings, it's based on the percentage of carcasses found that are bigfoot, the percentage of carcasses found that are not bigfoot, etc. I've summarized the horserace example in the link you originally posted as a comparison for what I have set up. Tell me what you think.

Summary of Kevin Boone Website:
Question: What is the probability that Dogmeat (a horse) will win the race if it’s raining?
Bayes’ Theorem: p(w|r) = p(r|w) * p(w) / p(r)
p = probability
w = win
r = rain
Formula: [probability of win given rain] = [probability of rain given win] * ([probability of win] / [probability of rain])

-----------------------Raining-----Not Raining
Dogmeat Wins------------3------------2
Dogmeat Does Not Win----1------------6

p(r|w) = 3/5 = 60%
p(w) = 5/12 = 41.7%
p(r) = 4/12 = 33.3%

p(w|r) = (3/5)*[(5/12) / (4/12)]
p(w|r) = 0.75 = 75%

-----------------------------------------------------------------------

Applied to Bigfoot:
Question: What is the probability that a bigfoot carcass will be found if bigfoot exists?
Bayes’ Theorem: p(c|b) = p(b|c) * p(c) / p(b)
p = probability
c = carcass
b = bigfoot
Formula: [probability of carcass given bigfoot] = [probability of bigfoot given carcass] * ([probability of carcass] / [probability of bigfoot])

-----------Bigfoot------Not Bigfoot
Carcass------0%---------100%
No Carcass---0%---------100%

Verbal explanation of the chart above:
1) Percentage of times there was a Carcass and it was Bigfoot = 0%
2) Percentage of times there was a Carcass and it was Not Bigfoot = 100%
3) Percentage of times there was No Carcass and it was Bigfoot = 0%
4) Percentage of times there was No Carcass and it was Not Bigfoot = 100%

0=0% , 1=100%

p(b|c) = 0/1 = 0%
p(c) = 1/1 = 100%
p(b) = 0/1 = 0%

p(c|b) = 0% * [100% / 0%] = (0%/0%) * 100%
p(c|b) = 100% (Zeros in numerator and denominator cancel out and/or remove what’s undefined from the previous step)

P1: The probability that a bigfoot carcass will be found if bigfoot exists is 100%.
P2: No bigfoot carcass has ever been found.
C: Therefore, based on current evidence, bigfoot does not exist.
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August 12th, 2015, 11:03 AM   #8
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Yes, except I'd adapt with some version of the sunrise problem bootstrapping to avoid giving a literal 0%. My preferred version (not the original) gives an extra 1/2 positive and an extra 1/2 negative observation.
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