Edge 2006
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The question of the year at the Edge (http://www.edge.org) is
"The history of science is replete with discoveries that were considered socially, morally, or emotionally dangerous in their time; the Copernican and Darwinian revolutions are the most obvious. What is your dangerous idea? An idea you think about (not necessarily one you originated) that is dangerous not because it is assumed to be false, but because it might be true?" Several people objected that they wouldn't contribute exactly because their ideas were dangerous ?_? I expect that different responses will appeal to each of us. For me, the response by Bart Kosko (http://www.edge.org/ q2006/q06_11.html), was the most interesting -- "Most bell curves have thick tails." In other words, he is worrying that statistical outliers may be more probable than predicted by the Gaussian normal distribution. Kosko focuses his discussion on the infinite set of possible stable bell curves that arise when you don't assume away infinite variance, and he claims that most of these curves have wider tails than the normal bell curve. I suspect he means "most" in some mathematical sense, and that he is worrying whether it is also true that most of the distributions we experience have fat tails. For example, whether quantum uncertainty is "fat." He mentions other bell curves like Laplace bell curves, but he doesn't mention power law curves specifically. But he doesn't explain enough of the math for me to understand whether he is including power law cases. Does anyone know enough about this to comment? -Roger |
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Kosko raises important points that often get neglected in practice. Yes, not
all bell-shaped distributions are normal. And yes, in practice an adequate probability model may require thicker tails than a normal distribution. Some of his contentions I would question, however: 1. Most distributions have infinite variance? This can't make sense. First, in practice, no random quantity can have infinite support so "the best of all possible" representations of real uncertainty cannot have infinite variance. Think about anything real--Google stock price on June 1, number of Internet hits to a site in 2006, the national debt in 2007--all of these are bounded so have finite support though on a perhaps wide interval. So even the normal distribution with its infinite support but finite variance uses infinite support as a simplifying feature of the probability model. So if all "real" distributions have finite variance the usual central limit theorem is valid. 2. He questions using variance as a measure of dispersion, but presents no alternatives, but hey there are a bundle of them like expected absolute deviations about the median, interquartile range, etc. Also he doesn't really give compelling reasons for not using the varaiance. 3. He at times seems to conflate thicker-tailed (than the normal) with infinite variance. Obviously not valid. 4. He neglects mixture models in spite of their success in representing thicker tailed distributions. Think about line voltage from an electrical socket. It might under usual conditions be normally distributions, but with some probability usual does not prevail and the voltage may come from a normal distribution with much larger variance. So overall we observe voltage from a mixture of the two normal distributions, and the mixture will have thicker tails than that of a single normal distribution. 5. Proper interpretation of the tests for outliers he criticizes is something like, "If the distribution were normal, these observations would be highly unusual". So logicially the test is as much a test of normality as it is one of identifying unusual points. 6. With due respect to the fun of Wikipedia, here's a reference that gives some pretty good information about the technical basis behind what Kosko discusses: http://en.wikipedia.org/wiki/Levy_alpha-stable_distributions George On 1/3/06, Roger Frye <rfrye at commodicast.com> wrote: > > The question of the year at the Edge (http://www.edge.org) is > > "The history of science is replete with discoveries that were considered > socially, morally, or emotionally dangerous in > their time; the Copernican and Darwinian revolutions are the most obvious. > What is your dangerous idea? An idea you > think about (not necessarily one you originated) that is dangerous not > because it is assumed to be false, but because > it might be true?" > > Several people objected that they wouldn't contribute exactly because > their ideas were dangerous ?_? > > I expect that different responses will appeal to each of us. For me, the > response by Bart Kosko (http://www.edge.org/ > q2006/q06_11.html), was the most interesting -- "Most bell curves have > thick tails." In other words, he is worrying > that statistical outliers may be more probable than predicted by the > Gaussian normal distribution. > > Kosko focuses his discussion on the infinite set of possible stable bell > curves that arise when you don't assume away > infinite variance, and he claims that most of these curves have wider > tails than the normal bell curve. I suspect he > means "most" in some mathematical sense, and that he is worrying whether > it is also true that most of the distributions > we experience have fat tails. For example, whether quantum uncertainty is > "fat." > > He mentions other bell curves like Laplace bell curves, but he doesn't > mention power law curves specifically. But he > doesn't explain enough of the math for me to understand whether he is > including power law cases. Does anyone know > enough about this to comment? > -Roger > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at Mission Cafe > lectures, archives, unsubscribe, maps at http://www.friam.org > -- George T. Duncan Professor of Statistics Heinz School of Public Policy and Management Carnegie Mellon University Pittsburgh, PA 15213 (412) 268-2172 -------------- next part -------------- An HTML attachment was scrubbed... URL: http://redfish.com/pipermail/friam_redfish.com/attachments/20060103/25295fe2/attachment.htm |
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In reply to this post by Roger Frye-2
> He mentions other bell curves like Laplace bell curves, but he doesn't
> mention power law curves specifically. But he > doesn't explain enough of the math for me to understand whether he is > including power law cases. Does anyone know > enough about this to comment? At last year's Lake Arrowhead Conference, Bill McKelvey gave quite an impassioned discussion on the shock to traditional statistics power laws will be, especially in the social sciences. He gave out a preliminary version of a paper he's working on and said it would be fine to distribute and get comments on. http://backspaces.net/files/PowerLaws.pdf Also, last year Rob Axtell gave a good talk that included a digression into this area which included a suite of "central limit theorems". I don't know if he's published yet however. Let us know if you find anything interesting. I particularly liked both takes on the impact on statistics they gave. Prospero A?o a todos! -- Owen Owen Densmore http://backspaces.net - http://redfish.com - http://friam.org On Jan 3, 2006, at 8:21 AM, Roger Frye wrote: > The question of the year at the Edge (http://www.edge.org) is > > "The history of science is replete with discoveries that were > considered > socially, morally, or emotionally dangerous in > their time; the Copernican and Darwinian revolutions are the most > obvious. > What is your dangerous idea? An idea you > think about (not necessarily one you originated) that is dangerous not > because it is assumed to be false, but because > it might be true?" > > Several people objected that they wouldn't contribute exactly because > their ideas were dangerous ?_? > > I expect that different responses will appeal to each of us. For > me, the > response by Bart Kosko (http://www.edge.org/ > q2006/q06_11.html), was the most interesting -- "Most bell curves have > thick tails." In other words, he is worrying > that statistical outliers may be more probable than predicted by the > Gaussian normal distribution. > > Kosko focuses his discussion on the infinite set of possible stable > bell > curves that arise when you don't assume away > infinite variance, and he claims that most of these curves have wider > tails than the normal bell curve. I suspect he > means "most" in some mathematical sense, and that he is worrying > whether > it is also true that most of the distributions > we experience have fat tails. For example, whether quantum > uncertainty is > "fat." > > He mentions other bell curves like Laplace bell curves, but he doesn't > mention power law curves specifically. But he > doesn't explain enough of the math for me to understand whether he is > including power law cases. Does anyone know > enough about this to comment? > -Roger > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at Mission Cafe > lectures, archives, unsubscribe, maps at http://www.friam.org |
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In reply to this post by George Duncan
Thank you, George. That was the kind of serious critique I was hoping for.
Note to the power-law geeks, the wiki article (http://en.wikipedia.org/wiki/Levy_alpha-stable_distributions) that George refers to claims "A generalization due to Gnedenko and Kolmogorov states that the sum of a number of random variables with power-law tail distributions decreasing as 1 / | x | ^(? + 1) (and therefore having infinite variance) will tend to a stable Levy distribution f(x;?,0,c,0) as the number of variables grows." -Roger |
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In reply to this post by Roger Frye-2
[Resend .. I didn't see it on the list. Sorry if you see it twice!]
> He mentions other bell curves like Laplace bell curves, but he doesn't > mention power law curves specifically. But he > doesn't explain enough of the math for me to understand whether he is > including power law cases. Does anyone know > enough about this to comment? At last year's Lake Arrowhead Conference, Bill McKelvey gave quite an impassioned discussion on the shock to traditional statistics power laws will be, especially in the social sciences. He gave out a preliminary version of a paper he's working on and said it would be fine to distribute and get comments on. http://backspaces.net/files/PowerLaws.pdf Also, last year Rob Axtell gave a good talk that included a digression into this area which included a suite of "central limit theorems". I don't know if he's published yet however. Let us know if you find anything interesting. I particularly liked both takes on the impact on statistics they gave. Prospero A?o a todos! -- Owen Owen Densmore http://backspaces.net - http://redfish.com - http://friam.org |
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Here's a slightly newer version of the McKelvey paper I mentioned
earlier: http://backspaces.net/files/BeyondGausian.pdf -- Owen Owen Densmore http://backspaces.net - http://redfish.com - http://friam.org Begin forwarded message: > From: Owen Densmore <owen at backspaces.net> > Date: January 3, 2006 11:37:31 AM MST > To: The Friday Morning Applied Complexity Coffee Group > <Friam at redfish.com> > Subject: Re: [FRIAM] Edge 2006 > > [Resend .. I didn't see it on the list. Sorry if you see it twice!] > >> He mentions other bell curves like Laplace bell curves, but he >> doesn't >> mention power law curves specifically. But he >> doesn't explain enough of the math for me to understand whether he is >> including power law cases. Does anyone know >> enough about this to comment? > > At last year's Lake Arrowhead Conference, Bill McKelvey gave quite > an impassioned discussion on the shock to traditional statistics > power laws will be, especially in the social sciences. He gave out > a preliminary version of a paper he's working on and said it would > be fine to distribute and get comments on. > http://backspaces.net/files/PowerLaws.pdf > > Also, last year Rob Axtell gave a good talk that included a > digression into this area which included a suite of "central limit > theorems". I don't know if he's published yet however. > > Let us know if you find anything interesting. I particularly liked > both takes on the impact on statistics they gave. > > Prospero A?o a todos! > > -- Owen > > Owen Densmore > http://backspaces.net - http://redfish.com - http://friam.org > > |
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