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Title Author Date
Variance and probability in Nilsson and Pelger Wein, Richard Sep 08, 2003
David Berlinski insists that Nilsson and Pelger's model "lacked any feature corresponding to random variations". But that is precisely the role played by the coefficient of variance in Falconer's formula, Berlinski's objections notwithstanding. Nilsson and Pelger's model assumes a population in which certain "quantitative characters" vary randomly within the population. In other words, these characters are random variables. Falconer's formula purports to tell us how much we should expect a character to change in one generation given that it is a random variable with a certain variance (and given certain other parameters). (By the way, I have not read Falconer's paper and so can make no comment on the validity of the formula. However Berlinski's point seems to be an objection not to Falconer's formula but to Nilsson and Pelger's use of it.) How then can Berlinski believe that the model "lacked any feature corresponding to random variations"? On the face of it, this claim seems absurd. I suspect the answer is that Berlinski is confusing variation within a population in a given generation with variation of a population across generations.

In Nilsson and Pelger's model, characters are random variables across a population. But the random behaviour of a character across the population gives rise to non-random "response", R, the change in the population mean from one generation to the next. If, in generation i, the character is a random variable with mean m_i, then in generation i+1 it is another random variable with mean m_i + R_i. (The coefficient of variance remains constant.) In other words, the probability distribution of a character changes over the generations, as the population evolves.

Now, in reality one would expect the change from generation to generation to be random too, so R should ideally be considered the mean of another random variable. However, since Nilsson and Pelger are summing over many generations, it does no harm for them to assume that the actual change in each generation will equal the mean (statistically expected) change for that generation. But perhaps this is the source of Berlinski's confusion. Perhaps he is mistaking the lack of random variation across generations for a lack of random variation within the population.

I hope this will help clarify matters. It should, for example, correct Berlinski's belief that the range of possible values of a character is fixed in Nilsson and Pelger's model. (He writes: "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.") There is no fixed "range of a random variable" in the model, because the probability distribution is assumed to change over the generations. The model imposes no limits on the range of a character.
Related Articles: Has Darwin met his match in Berlinski?

Title Author Date
Variance and probability in Nilsson and Pelger Berlinski, David Sep 22, 2003
Part I
In his posting of September 8th, 2003, Erik (no last name given) lists seven points. I comment in turn:
1) It is not possible to respond rationally to the charge that some of my claims are strange. I would be happy to discuss the point off-line.
2) I used the historical example of womens shoe sizes to point to a set of circumstances in which the distribution of variance within a sample population  its initial probability distribution  conveys no clue whatsoever concerning the underlying dynamics of change between sample populations  their probability transition system. The point is obvious. In the case of women's shoe sizes, the mean value of a quantitative trait is governed by a deterministic and not a stochastic process. It is worth noting that if producers guess at the future mean value of the price of a given commodity, then by a theorem of Samuelson it follows that future prices of that commodity will describe a random walk. In my original Commentary articles, I argued that whatever else it might be, Nilsson and Pelger's paper failed signally to incorporate a key Darwinian assumption  that of random variation. A number of contributors to Talk Reason  Downard and Wein, for example  are persuaded that I am mistaken. I elaborate and make precise the point in response to Richard Wein.
3) Nilsson and Pelger's model is not accurate until its end point. Whatever could the contrary claim mean? Nilsson and Pelger's model is continuous, so errors in the model accumulate continuously as well.
4) It is possible to rephrase my ideas here without specifying a bijection; details tend to become messy, as Erik observes.
5) I do not understand this point and so cannot comment.
6) I am at a loss to know what Erik might mean in affirming that "population genetics wasn't formulated to prove that evolution is possible." Of course it was. And not only possible, but true as well. That is precisely the point of mathematical population genetics. My question is whether population genetics comprises a classical example of a theory with so many free parameters that a genuine confrontation with experience becomes impossible.
7) "Unlike creationists," Erik writes, "biologists do not attempt to read off the history of an organ by studying it in isolation." I invite Erik to consider Nilsson and Pelger's paper as a counter-example.
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Title Author Date
Variance and probability in Nilsson and Pelger Berlinski, David Sep 22, 2003
Part II
Richard Wein argues that in having cited John Woodmorappe's paper, I have reposed my confidence in an argument from innuendo. In precisely the same context in which I cited Woodmorappe, I cited journal articles in which my own position on the Cambrian explosion, and its significance for Darwinian theory, was held open to challenge. The details are available on-line at the Discovery Institute's web site. In neither case did I endorse criticisms of my own views: I simply cited them. Richard Wein has discerned no innuendo in the second set of citations that I presented. I would ask why not? The fact of the matter is that scientists and scholars regularly cite articles with which they do not agree or which they do not find compelling. What of it? I agree with James Downard that this policy has its limits, and I said so in responding to his letter. It is irresponsible, and pointless, to cite work that is plainly preposterous. I do not believe Woodmorappe's paper has met this standard. Finally, Richard Wein asks that I supply a single reference in which I defend the reptile to mammal sequence. Easy enough. My original Commentary article referred to the sequence as the jewel in the crown of Darwinian paleontology. Defensive enough?
Related Articles: Has Darwin met his match in Berlinski?

Title Author Date
Variance and probability in Nilsson and Pelger Berlinski, David Sep 22, 2003
Part III
Richard Wein is persuaded that I have committed a great absurdity in insisting that Nilsson and Pelger's paper contains nothing corresponding to the requisite random variations that are called for in Darwinian theory. He is quite mistaken. What is at issue is whether changes in a population from one generation to another arise as the expression of an underlying stochastic process. It is, of course, the only point at issue. If the dynamics of change are not essentially stochastic, the underlying theory is not Darwinian. I take it that this is by definition what a Darwinian theory requires. In their work, Nilsson and Pelger do specify an initial probability distribution over a sample population. They never provide the requisite probability transition system. Their dynamical model is entirely deterministic.
Nilsson and Pelger's results are presented in two stages. In the first, Nilsson and Pelger investigate the fate of a single light sensitive cell. Nilsson and Pelger assume that changes to their initial patch are governed by the function
1) f(a)n

subject to the boundary condition that for a = 1.01, f(a)n = 80, 129, 540, whence n = 1829.
1), of course, is simply a stripped-down formula for the generation of compound interest. 1) determines an action f --> f --> ... --> f, such that the transition probability Pr(fi fi+1) = 1 for any two actions of the function. The point is again obvious, indeed, trivial. No Darwinian assumptions are at work; no stochastic features of any sort.
In the second stage of their paper, Nilsson and Pelger discount a by means of Falconer's short term response statistic R. Short term, note. Falconer's response statistic is not meant to model the long term behavior of a sample population. The discounted value of a now stands for the mean value MV of visual acuity in every generation.
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Title Author Date
Variance and probability in Nilsson and Pelger Berlinski, David Sep 22, 2003
Part IIIa
Let us now normalize Nilsson and Pelger's dynamic equation by setting coefficients of heredity and selection to 1. Neither selection nor heredity address the source of dynamical change. Their dynamical theory is again expressed as a function
2) f(MVINITIAL)n.

But since MVINITIAL = 1.01, when coefficients of heredity and selection are 1, 2) is nothing more than a restatement of 1), subject, in fact, to precisely the same boundary condition: f(MVINITIAL)n = 80, 129, 540, whence n = 1829 again.
It follows again that the transition probability for each fi fi+1 = 1. No surprise, this.
To regard this as a Darwinian model, or to imagine that 2) specifies a probability transition system, is simply a mistake.
What might reasonable transition probabilities look like? I have no idea, of course, but there is absolutely no reason to suppose that the MV will rise monotonically throughout the course of 330,000 generations. It may well decline, leading to negative selection and a net reduction of population, or it may oscillate persistently around the initial mean. But one thing is clear. Nilsson and Pelger's model requires that an emerging eye undergo 1829 consecutive positive one percent changes. If each one percent step corresponds to a mutation, I would conservatively estimate that the chances of observing 1829 consecutive positive mutations in precisely the same genetic locus is effectively zero. So would everyone else.
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Title Author Date
Variance and probability in Nilsson and Pelger Berlinski, David Sep 22, 2003
Part IV
In my Commentary article I observed that Nilsson and Pelger provided no calculations justifying their central claim, namely that 1829 one percent steps are sufficient to transform a light sensitive patch into a camera eye. The list that Mr. Curtis offers is of scant help in this regard. More specifically,
1 It is incomplete
2 It is largely incomprehensible. What, for example, does "corneal width (curve) 46.5 46.50" mean?
3 It is irrelevant inasmuch as it was not included in Nilsson and Pelger's original paper nor in their notes.
And, finally
4 It is absurd, inasmuch as a list is not a calculation. In my essay, I asked how Nilsson and Pelger's numbers were derived? They do not say and neither does Mr. Curtis. Suppose, for example, that Nilsson and Pelger's original light sensitive patch were to increase in length, and length alone, by 1829 one percent steps. Would that result in a structure similar to the one they derive, or would it be different? In either case, how would one know, without a specification of overall morphological change by means of a single derived unit of morphological change amalgamating all the dimensions of change? A very rich literature now exists dealing with the metric structure of complex three-dimensional biological objects. References are available on line under my name at the Discovery Institute web site.
One final point. I have never accused Nilsson and Pelger of fraud. I consider their paper absurd, but that is another matter entirely. The fraud in question involves the misrepresentation of their work, chiefly but not only by Richard Dawkins.
Related Articles: Has Darwin met his match in Berlinski?

Title Author Date
Variance and probability in Nilsson and Pelger Wein, Richard Sep 23, 2003
David Berlinski writes: "What is at issue is whether changes in a population from one generation to another arise as the expression of an underlying stochastic process. It is, of course, the only point at issue."

On the contrary, it is not at all the point at issue! Nilsson and Pelger take for granted the fact that such changes can arise. The justification for that assumption is outside the scope of their paper. The issue they address is whether such changes can accumulate to produce an eye, and how long such a process is likely to take.

Berlinski appears to believe that a stochastic process cannot be represented by a deterministic model. If so, he is clearly wrong. Deterministic algorithms are often used to model stochastic processes. Does a casino manager use cards, dice or other randomizers in estimating his future takings? I doubt it. Due to the large number of random events, he makes the approximating assumption that his takings will be in accordance with the statistical expectation, which he calculates by means of a purely deterministic algorithm. Similarly, Nilsson and Pelger make the approximating assumption that the quantitative change in each generation will be the statistically expectated value estimated by Falconer's formula.

Berlinski is too concerned with whether Nilsson and Pelger's model can be labelled "Darwinian". I'm not even sure what he means by this. If he means that their algorithm is not Darwinian, then I agree. But just as one can use a non-stochastic algorithm to model a stochastic process, one can use a non-Darwinian algorithm to model a Darwinian process. Berlinski is chasing a red herring. The real issue is not how we label the model, but whether the assumptions and approximations it makes are justifiable.

His failure to appreciate that a model is an abstraction leads Berlinski into further blunders: "...there is absolutely no reason to suppose that the MV will rise monotonically throughout the course of 330,000 generations." Indeed, but the model does not require that it do so. It merely assumes that it will do so for the purposes of approximation. All the model reqires is that the mean value rises on average by roughly the calculated amount. Sometimes it may fall, other times it may rise faster and catch up.

[Regarding my allegation of argument by innuendo, I will respond to Berlinski in the column where I originally made that allegation (titled "James Downard").]
Related Articles: Has Darwin met his match in Berlinski?