Suppose the chance of finding a tiger somewhere in a given household, on a given day, is one in a billion.
Or so say the pro-tigerians. The tiger denialist faction, of course, claims that statistic is made-up, and tigers don’t actually exist. But one household in a trillion might hallucinate a tiger, on any given day.
Today, you search your entire house—the dishwasher AND the fridge AND the trashcan etc. P(You find a tiger|tigers exist) = .000000001 P(You don’t find a tiger|tigers don’t exist) = .000000000001 P(You don’t find a tiger|tigers exist) = .999999999 P(You don’t find a tiger|tigers don’t exist) = .999999999999
And suppose you are 99.9% confident that tigers exist—you think you could make statements like that a thousand times in a row, and be wrong only once. (Perhaps rattling off all the animals you know.) Your prior odds ratio is 999 to 1.
So you take your prior odds, (.999/.001) and multiply by the likelihood ratio, (.999999999/.999999999999), to get a posterior odds ratio of 998.999999002 to 1. This is, clearly, a VERY small adjustment.
What if you search more households: how many would you have to search, without finding a tiger, before you dropped just to 90% confidence in tigers, where you still think tigers exist but would not willingly bet your life on it? If I’ve done the math right, about five billion. There probably aren’t that many households in the world, so searching every house would be insufficient to get you down to just 90% confidence, much less 10% or whatever threshold you’d like to use for “tigers probably don’t exist”.
(And my one-in-a-billion figure is probably far too high, and so searching every household in the world should get you even less adjustment...)
But if you could search a trillion houses at those odds, and still never found a tiger—then you’d be insane to still claim that tigers probably do exist.
And if a trillion searches can produce such a shift, then each individual search can’t produce no evidence. Just very little.
Suppose the chance of finding a tiger somewhere in a given household, on a given day, is one in a billion. Or so say the pro-tigerians. The tiger denialist faction, of course, claims that statistic is made-up, and tigers don’t actually exist. But one household in a trillion might hallucinate a tiger, on any given day.
Today, you search your entire house—the dishwasher AND the fridge AND the trashcan etc.
P(You find a tiger|tigers exist) = .000000001
P(You don’t find a tiger|tigers don’t exist) = .000000000001
P(You don’t find a tiger|tigers exist) = .999999999
P(You don’t find a tiger|tigers don’t exist) = .999999999999
And suppose you are 99.9% confident that tigers exist—you think you could make statements like that a thousand times in a row, and be wrong only once. (Perhaps rattling off all the animals you know.) Your prior odds ratio is 999 to 1. So you take your prior odds, (.999/.001) and multiply by the likelihood ratio, (.999999999/.999999999999), to get a posterior odds ratio of 998.999999002 to 1. This is, clearly, a VERY small adjustment.
What if you search more households: how many would you have to search, without finding a tiger, before you dropped just to 90% confidence in tigers, where you still think tigers exist but would not willingly bet your life on it? If I’ve done the math right, about five billion. There probably aren’t that many households in the world, so searching every house would be insufficient to get you down to just 90% confidence, much less 10% or whatever threshold you’d like to use for “tigers probably don’t exist”.
(And my one-in-a-billion figure is probably far too high, and so searching every household in the world should get you even less adjustment...)
But if you could search a trillion houses at those odds, and still never found a tiger—then you’d be insane to still claim that tigers probably do exist.
And if a trillion searches can produce such a shift, then each individual search can’t produce no evidence. Just very little.
I’ve posted a comment that answers you here