6 ms·

With fat-tailed distributions, increasing the sample size actually makes it more likely that you'll get one element much greater than the others.

TreeRingCounter 4y ago
This sounds like nonsense. We're talking about multivariate distributions, and you haven't defined a norm by which ordinal comparisons can be made between sample points.
- brewmarche 4y ago
  All norms on finite-dimensional vector spaces are equivalent so the choice of norm shouldn’t matter for tails really.
  - hgsgm 4y ago
    Norms are equivalent in the sense they differ by less than a constant in both directions, not in the sense that outliers are the same in all norms.
    - brewmarche 4y ago
      Norms are for dealing with magnitude not direction (and were brought up by my parent commenter). If you care about direction specify an angle, quadrant, cone or other subregion that allows you to take the limit to infinity which then doesn’t depend on the norm. Note this is the same in the univariate case where we talk about left and right tail if we need to distinguish.
  - TreeRingCounter 4y ago
    > All norms on finite-dimensional vector spaces are equivalent How are you going to tell me that an L1 norm is equivalent to an L2 norm, for example?
    - brewmarche 4y ago
      This is a standard result which you can look up in most text books, or here on StackExchange: https://math.stackexchange.com/questions/57686/understanding-of-the-theorem-that-all-norms-are-equivalent-in-finite-dimensional https://math.stackexchange.com/questions/57686/understanding... In the end it doesn’t whether we go to infinity in one norm or the other. Note that I am talking about finite dimensions, so I guess you didn’t mean the L^p norms or \ell^p for integrable functions or sequences but the finite-dimensional p-norms.
      - TreeRingCounter 4y ago
        This theorem is completely irrelevant - the equivalence relation described by the theorem does not imply an equivalence between ordinal relationships imposed by different choice of norm. Also, "tails" isn't the same as "at infinity".
jonahx 4y ago
Isn't the surprising part that it's fat-tailed rather than normal?
- eloff 4y ago
  Why would it be? All careers with leverage seem to follow a Pareto distribution. It would surprise me if any sport, art, or creative endeavor was normally distributed.
  - jonahx 4y ago
    You're right. I was thinking the "initial talent" might be normal in the way IQ scores are, but the parent was talking about actual performance, where history, training, etc all come into play. And empirically, you have Gretzky in hockey, Jordan in basketball....
    - OscarCunningham 4y ago
      IQ is normally distributed because it's graded on a curve to make that true by definition. I'm not sure how it's distributed on any sort of 'natural' scale.
      - jacquesm 4y ago
        What would such a natural scale look like?
        derefr 4y ago
        Maybe a cross-species IQ score would be interesting. Chimpanzees and corvids both can solve many IQ-test problems, but also are not experiencing the Flynn effect as far as we know. So we could measure IQ in multiples of a chimp's IQ, just like we measure cars in multiple's of a workhorse's sustainable pulling output. (Which is actually a less objective measure, when you think about it; we were breeding workhorses to increase that very target at the time the concept of "horsepower" was invented!)
SmooL 4y ago
Yes, but if you have many fat tailed distributions, increasing the sample size makes it increasingly less likely that it's _the same person at the end each time_
- OscarCunningham 4y ago
  This would imply that there's a single factor upstream leading to his exceptional ability in several measures downstream.
  - cryptonector 4y ago
    https://medium.com/@casciari/messi-is-a-dog-18b8f32766c8 https://medium.com/@casciari/messi-is-a-dog-18b8f32766c8 https://www.youtube.com/watch?v=mDsztJOgqUg https://www.youtube.com/watch?v=mDsztJOgqUg TL;DR / TL;DW: Messi is a dog. The ball is his.
    - ByThyGrace 4y ago
      Recommended. For further context, it's the narration of an article written in Spanish by H. Casciari, translated to English. It's a lovely but also profound literary article in the subject of Messi never losing sight of the ball, while also a critique on modern day analysis of the game and its influence on player behavior.
      - cryptonector 4y ago
        In Spanish: https://www.youtube.com/watch?v=Mg9fDOvE80g https://www.youtube.com/watch?v=Mg9fDOvE80g

TreeRingCounter 4y ago

This sounds like nonsense. We're talking about multivariate distributions, and you haven't defined a norm by which ordinal comparisons can be made between sample points.

brewmarche 4y ago
All norms on finite-dimensional vector spaces are equivalent so the choice of norm shouldn’t matter for tails really.
- hgsgm 4y ago
  Norms are equivalent in the sense they differ by less than a constant in both directions, not in the sense that outliers are the same in all norms.
  - brewmarche 4y ago
    Norms are for dealing with magnitude not direction (and were brought up by my parent commenter). If you care about direction specify an angle, quadrant, cone or other subregion that allows you to take the limit to infinity which then doesn’t depend on the norm. Note this is the same in the univariate case where we talk about left and right tail if we need to distinguish.
- TreeRingCounter 4y ago
  > All norms on finite-dimensional vector spaces are equivalent How are you going to tell me that an L1 norm is equivalent to an L2 norm, for example?
  - brewmarche 4y ago
    This is a standard result which you can look up in most text books, or here on StackExchange: https://math.stackexchange.com/questions/57686/understanding-of-the-theorem-that-all-norms-are-equivalent-in-finite-dimensional https://math.stackexchange.com/questions/57686/understanding... In the end it doesn’t whether we go to infinity in one norm or the other. Note that I am talking about finite dimensions, so I guess you didn’t mean the L^p norms or \ell^p for integrable functions or sequences but the finite-dimensional p-norms.
    - TreeRingCounter 4y ago
      This theorem is completely irrelevant - the equivalence relation described by the theorem does not imply an equivalence between ordinal relationships imposed by different choice of norm. Also, "tails" isn't the same as "at infinity".

jonahx 4y ago

Isn't the surprising part that it's fat-tailed rather than normal?

eloff 4y ago
Why would it be? All careers with leverage seem to follow a Pareto distribution. It would surprise me if any sport, art, or creative endeavor was normally distributed.
- jonahx 4y ago
  You're right. I was thinking the "initial talent" might be normal in the way IQ scores are, but the parent was talking about actual performance, where history, training, etc all come into play. And empirically, you have Gretzky in hockey, Jordan in basketball....
  - OscarCunningham 4y ago
    IQ is normally distributed because it's graded on a curve to make that true by definition. I'm not sure how it's distributed on any sort of 'natural' scale.
    - jacquesm 4y ago
      What would such a natural scale look like?
      - derefr 4y ago
        Maybe a cross-species IQ score would be interesting. Chimpanzees and corvids both can solve many IQ-test problems, but also are not experiencing the Flynn effect as far as we know. So we could measure IQ in multiples of a chimp's IQ, just like we measure cars in multiple's of a workhorse's sustainable pulling output. (Which is actually a less objective measure, when you think about it; we were breeding workhorses to increase that very target at the time the concept of "horsepower" was invented!)

SmooL 4y ago

Yes, but if you have many fat tailed distributions, increasing the sample size makes it increasingly less likely that it's _the same person at the end each time_

OscarCunningham 4y ago
This would imply that there's a single factor upstream leading to his exceptional ability in several measures downstream.
- cryptonector 4y ago
  https://medium.com/@casciari/messi-is-a-dog-18b8f32766c8 https://medium.com/@casciari/messi-is-a-dog-18b8f32766c8 https://www.youtube.com/watch?v=mDsztJOgqUg https://www.youtube.com/watch?v=mDsztJOgqUg TL;DR / TL;DW: Messi is a dog. The ball is his.
  - ByThyGrace 4y ago
    Recommended. For further context, it's the narration of an article written in Spanish by H. Casciari, translated to English. It's a lovely but also profound literary article in the subject of Messi never losing sight of the ball, while also a critique on modern day analysis of the game and its influence on player behavior.
    - cryptonector 4y ago
      In Spanish: https://www.youtube.com/watch?v=Mg9fDOvE80g https://www.youtube.com/watch?v=Mg9fDOvE80g