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# Discussion

Title Author Date
Variance and probability in Nilsson and Pelger Berlinski, David Aug 23, 2003
Part I
In my Commentary essays, I argued that Nilsson and Pelger's theory, whatever else it might be, was certainly not Darwinian inasmuch as it lacked any feature corresponding to random variations. Mr. Downard disagrees. The fact that Nilsson and Pelger specified their coefficient of variance within a narrow range, he has argued, is sufficient to establish the Darwinian nature of their theory. The point is of great importance. If variations are not random, Darwin's theory is liable to a regress in which explaining the source of those variations becomes the very problem that Darwin's theory was intended to solve.

Although the existence of variations within (or between) populations is necessary to establish an underlying stochastic structure, it is hardly sufficient. The distribution of women’s shoe sizes has historically displayed a tightly bounded variation around a mean, but as women's feet have gotten larger, so, too, women's shoes, and so, too, the variations themselves. No one suggests that the fact that there is and has always been variations in the size of women's shoes implies that these variations are random. Within Nilsson and Pelger’s theory, moreover, the coefficient of variance - the ratio of the standard deviation to the mean - is obviously correlated to the mean itself, and since the mean rises in each generation by a constant factor by Falconer’s response statistic R, the coefficient of variance can hardly be considered as the expression of some random variable. Finally, since the mean rises monotonically throughout 330,000 generations, and since the coefficient of variance remains a constant percentage of the mean, new variations must be introduced into their sample population in each generation if variations are not to shrink to zero. Nilsson and Pelger say nothing about the source of these new variations, either in terms of mutations or sampling errors. But then, of course, if new variations lie in the range of a random variable, the mean itself could not possibly arise by a fixed percentage in each generation. Finally, it is easy enough to derive a contradiction from Nilsson and Pelger’s use of Falconer's response statistic R as the basis of a recurrence relationship. Thus assume, as Nilsson and Pelger do, that there is a fixed upper bound UB to visual acuity beyond which the mean m cannot rise. As their population approaches UB with respect to visual acuity, variations must shrink to the right of the mean until UB = m. At this point, or shortly thereafter, the true mean and the statistical mean will coincide and the population will exhibit no variations beyond sampling errors. From this it follows that the coefficient of variation cannot always be a constant percentage of a rising mean.

Title Author Date
How about it? Berlinski, David Dec 22, 2002
Gentlemen:

I encourage you to submit your superb letters of remonstration to Commentary. I am quite certain, despite your doubts, that the magazine will find the space to publish them. If not, then I would suggest that you give me space in your on-line forum to respond directly.