Gauss’ Ghost
| March 17, 2010 | Posted by Robert Sheldon under Intelligent Design |
Johann Carl Friedrich Gauss was a polymath of no mean skill. Mathematicians bemoan the fact that he spent his later years doing physics, and physicists wish he had started earlier. One of his contributions was the derivation and proofs for the bell-shaped curve known as a “Gaussian” or “normal” distribution. It is the result of a random process in which small steps are taken in any direction. So universal is the “Gaussian” in all areas of life that it is taken to be prima facie evidence of a random process.
Only in recent years have people addressed situations that can deviate from a Gaussian. For example, one of the criteria that produce a Gaussian, is that the probability of a “small” step must be greater than the probability of a “big” step. That is, if we consider the random walk of the proverbial drunk near a lightpole, if he staggers in small steps most of the time, a plot of his position taken, say, every other second, would be a Gaussian. But if he staggers in big steps with a few small ones thrown in, then the plot begins to look peculiar. Instead of being a Gaussian, it develops a “fat tail“, with many locations far from the lightpole.
Now why is this important? Because many people predict that Darwinian evolution is driven by random processes of small steps. This implies that there must be some Gaussians there if we knew where to look.
33 Responses to Gauss’ Ghost
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Heinrich,
I am enlightened. And no, I rarely use Wikipedia, esp. in regard to ID
If you read my 1st comment, you would see the connection between intermittency and spatial diffusion. Which of course, is the weak point in the argument, but on the other hand, is often assumed.
You may be right about the exponential having a well behaved Diffusion coefficient, I’m afraid that my classes only discussed power laws, and I expanded the exponential as 1 – x + x2/2! …
and concluded it had a power law -1, which would give an infinite diffusion coeff. If there is a more elegant way to do it, I will have to read up on it.
As for the definition of “fat tails”, I was using it evidently in a non-mathematical way, for distributions whose tails have a smaller power law than Gaussian. Clearly not the Wikipedia definition.
But the point of the post was to consider what Pagel had discovered, which I think has been admirably accomplished.
@Robert Sheldong:
And no, I rarely use Wikipedia, esp. in regard to ID
*LOL* One got another impression, as your article contains at least four links to wikipedia. Only that your link for “fat tails” isn’t directed to their article on this subject, but to the one on the Lévy distribution…
#33
But the point of the post was to consider what Pagel had discovered, which I think has been admirably accomplished.
Robert – oddly at the end of it all I am not sure what you think he discovered. I think he found some evidence that a lot more speciation is the result of single events (rather than accumulated change) then was previously thought.