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Originally Posted by grant hutchison
Or to abandon it as an argument that's going nowhere, to be frank.
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I hope we won't have to do that, the answer is out there!
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Originally Posted by grant hutchison
(A paediatrician tells you that your son's height is in the 98th centile for his age. You go home and measure your son's height - he's 120cm tall. That immediately tells you that 2% of kids your son's age are over 120cm tall. You don't need to doubt that proportion just because you've now measured the height. It's the same proportion, with or without the numerical knowledge.)
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Your example is perfect, it shows us exactly what we need! But you have to change it into a probability argument. Let's say my son's height is 120cm. Can I assert with 95% confidence that he is above the 5% rank with no additional information? Yes I can, because I never used the 120cm, I can always make that assertion. The real question is, can I put limits on the median height at this same confidence, by using that 120cm? No, that is exactly what I can
not do! Here is how it would go. I'd assume height was a Gaussian variable, and I'd say I was 95% confident my son was above the 5% point. Let's say that means he is above half the median height (we'd need the standard deviation to allow this to be done, I'm picking numbers to come out simple). Can I now put a 95% confidence that the median height is below 240cm? There is a testable prediction, a calculation we can actually do and see who is right. I will wager anything reasonable that if we do this calculation, we will find that if we randomly select heights, and range over all possibilities, we will not find that 95% of the parents in our hypothetical study will be correct in the 95% limit they place on the median height! I don't know if it will be above or below 95%, but there's no reason to expect 95% as you must be claiming. (Unless, that is, there is something very special about Gaussian distributions, so I wouldn't wager too much until I'd thought that part out!)
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Originally Posted by grant hutchison
The set of relevance is the set of humans who make a claim about the likely total number of human lives, using their birth-number to make the calculation. 5% of them are wrong.
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Here you are ranging over all the people who make the calculation, so you are correct that 5% will be wrong. But we are not using the 5% number after ranging over all humans, we are using it just for ourselves, in saying that humanity is 5% likely to outlive 200 billion. The argument cannot be stated without the 200 billion number, so it has not ranged over all humans.
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Originally Posted by grant hutchison
Are you going to claim that there is a set of children 120cm high who are not in the 98th centile for their age, and so measuring your child's height invalidates the paediatrician's assessment?
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No, I'm not bringing in another parameter like age. I'm saying that the median height limit, for that age, cannot be calculated to 95% confidence using the single 120cm measurement (and the standard deviation information), because humans have an average height, and 120cm has some unkwown relationship to that height, so I can't make any confidence calculation about that average height.
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Originally Posted by grant hutchison
Here's another way of looking at it. Saying you're 95% percent certain you're not in the first 5% of human lives is just shorthand for saying "If only 100 humans lived, I'm 95% certain I wouldn't be in the first 5." Inserting your birthnumber merely introduces a constant of proportionality into that claim. Each individual inserts their own birth number, and comes up with their own estimate of the total number. The claim stays the same because the proportion is the same, just as it would have done if you'd decided to express your claim as a fraction or a per-mil, rather than a percent.
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The point is, since the number 200 billion appears in the Carter catastrophe argument, you have not ranged over all humans in presenting that argument, and so the 95% confidence
does not apply.