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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.
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Title Author Date
Variance and probability in Nilsson and Pelger Berlinski, David Aug 23, 2003
Part II
Consider Nilsson and Pelger’s light sensitive cells from two perspectives: first, as collection of combinatorial objects M endowed with an intrinsic metric dM, and, second, as a collection of proto-organs N in formation, these, too, endowed with their own, but quite separate metric dN. Distance in M is a measure of nearness in structure; distance in N a measure of difference in organisms with respect to visual acuity. Points in M are labeled t1, t2, ... , tn, and points in N, e1, e2, ..., ek. The function f: M --> N is a bijection. A probability transition system PR determines for each point t in M the probability that it will change to t’, together with an initial probability distribution PR0. The 4-tuple <M, N, f, PR> comprises a generic Darwinian structure.

In addition to the natural metric on M, there is an induced metric defined by the relationship: dN(I)(f-1e, f-1e’) = dN (e, e’).

We now perform a thought experiment. A distinguished element e* in N is fixed. A point t0 is selected in accordance with the initial probability distribution. The distance dN(0) from f(t0) is measured and the system engaged for i = 1, 2, 3, ... . As ti-1 moves to ti the distance dN(i) between f(ti) and e* is recorded.

The outcome of this experiment is the sequence {dN(i)} = dN(0), dN(1), ... , dN(n).

If f, PR> is a generic Darwinian structure, then <M, N, f, PR, {dN(i)}> is a generic Darwinian experiment. A Darwinian experiment is successful if for dN(0) at an average distance from e* the sequence {dN(i)} converges to a neighborhood of zero.
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Title Author Date
Variance and probability in Nilsson and Pelger Berlinski, David Aug 24, 2003
Part III

It is a curious fact that beyond biology there would seem to be no successful Darwinian experiments. And for obvious reasons. A Darwinian experiment establishes that {dN(i)} is both stable and oriented. The set of stable and oriented trajectories in N is typically of measure zero.

What must be added to a generic Darwinian structure to achieve a successful Darwinian experiment? The answer conveyed by the neo-Darwinian synthesis is that induced and natural metrics over M be correlated.

It is thus that mathematical population genetics introduces a new space S between M and N. S typically has the structure of a Euclidean vector space, and with mathematical population genetics S is carefully constructed to establish that dN(I)(f-1e, f-1e’) = dN (e, e') = dM (t, t’). A generic Darwinian structure is now defined as <M, N, f, PR, {dN(i)}, S >. Given such modified Darwinian structures, it is clear that Darwinian experiments are apt very often to be successful.

In Nilsson and Pelger’s paper, Falconer’s response statistic stands in for S, with the coefficient of variance, and constants for selection and heredity, adjusted to guarantee an outcome qualitatively in accord with their expectations. Under these conditions, how could one ever determine that so extraordinary an organ as the camera eye did not develop as the result of selection at all, but had its origins in completely different processes?
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Title Author Date
Variance and probability in Nilsson and Pelger Matzke, Nicholas Aug 24, 2003
Berlinski writes,

=====
In Nilsson and Pelger's paper, Falconer's response statistic stands in for S, with the coefficient of variance, and constants for selection and heredity, adjusted to guarantee an outcome qualitatively in accord with their expectations. Under these conditions, how could one ever determine that so extraordinary an organ as the camera eye did not develop as the result of selection at all, but had its origins in completely different processes?
=====

Berlinski needs to go back and re-read the N-P paper. It has three parts, which Berlinski repeatedly confuses with each other:

1) A determination that a sequence of eye types exists that is linked to each other by gradual improvements. *This*, and not any other section, is what determines that a gradual Darwinian path exists for the evolution of the eye.

2) An approximate *measurement* of how much change the above sequence takes. E.g., they measure changes in length, width, etc., of the various structures. They combine these measurements by counting the number of 1% changes that are required, getting a result of 1829 1% steps, if I recall correctly.

3) Then, an only then, do they address *how long* the above series of changes would take. They make *pessimistic* assumptions about heritability, variance, selection coefficient, etc., and using basic quantitative genetics, attain their result of a few hundred thousand years.

Berlinski appears to have a beef with the fact that the distribution of variation about the mean is assumed to remain symmetrical as the mean adjusts, but he fails to take into account the small value of that variance, 0.01, and the very small rate of change, 0.01%/generation if I recall correctly. This is much smaller than observed rates of change in natural populations, and the variance is smaller than the variance of quantitative traits in natural populations, e.g. height. It is therefore reasonable to assume that the mutation rate is sufficient to "keep up" the variability about the mean as it adjusts.
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Title Author Date
Variance and probability in Nilsson and Pelger Erik Sep 08, 2003
1. The connection between Darwin's theory and the pigeon-hole principle is strange and seems wrong, although at the same time not very important.
2. What is the point of the women's shoe size example? Is Berlinski trying to suggest that we shouldn't use coarse-grained stochastic models for variation in traits? Is Berlinski trying to suggest that variation in traits may not be stochastic in some ontological sense?
3. Berlinski's attempt to derive a contradiction from the assumption that the mean increases to a maximum value displays a misunderstanding of mathematical idealizations and approximations. If I were to assume that the
length of adult women's feet is a normally distributed stochastic variable, I'm sure Berlinski would accuse me of using a contradictory model, since an exact normal distribution would assign a non-zero probability to feet of negative length. However, it can still be a reasonable approximation.
Berlinski's actual complaint seems analogous. What matters is that Nilsson & Pelger's model is accurate up until the end point, since the end point itself contributes very little to total time required for the optics of an
eye to evolve.
4. If M is supposed to be the genotype space* and N is supposed to be some "organ space", then the mapping f:M-->N shouldn't be assumed to be a bijection, since we know that neutrality is a major feature of the genotype-to-phenotype map. (The choice of a bijective f is probably made to allow for a convenient definition of a metric on N in terms of a metric on M.) More can be said about his definition of a "generic Darwinian structure", but it's mostly about details.
5. The concept of a "successful Darwinian experiment" is a reasonable idealization, but it must be noted that it is not reasonable to understand it as defining exactly what we would consider to be a successful theoretical or empirical demonstration of the evolution of a trait. The requirement that the (induced) distance to some fixed point in "organ space" converges means that the evolutionary dynamics must stay in the same place in "organ space" forever once it is reached. Specialized to the case of eyes, it would mean that eyes couldn't change once they evolved, which can be a reasonable condition to impose on a model. But, as moles and cave-fish
will testify, it cannot be regarded as an exact and general condition. The attempt to make a point of the fact that the set of all mathematicallypossible convergent sequences is of measure zero indicates a failure to
understand which properties of the mathematical idealization that areaccurate and which are not. Since the real-world phenomenon modelled by the concept of a "successful Darwinian experiment" is not literally of infinite
duration in time, it is clearly unreasonable to attach such significance to convergence.
6. The last part of Berlinski's letter is a complaint that, by introducing a Euclidean vector space S and mapping elements from M to N via S, population geneticists guarantees that the fitness increases the closer to a distinguished point in "organ space" we get. This is perhaps true, but populations genetics wasn't formulated to prove that evolution is possible any more than the theory of electromagnetism was formulated to prove that
radio communication is possible. It was formulated to provide a description
of evolution and it is based on assumptions. And by studying modern populations, population geneticists can test these assumptions.
7. The letter ends by inquiring about under what conditions the "generic Darwinian structure" can be regarded as an unacceptable description of the facts. The concept is actually a little too general, because, e.g.,
Lamarckian models are included in the class of "generic Darwinian structure". Ignoring such factors, the answer to Berlinski's question is
that the most general Darwinian model contains so many free parameters that it is hard to think of a realistic way to test it just by looking at a particular organ in isolation. Unlike creationists, however, biologists do not attempt to read off the history of an organ by studying it isolation.
Instead, they study the genetics of organs, the reproductive advantage of organs, and patterns across species and across time. Such data do have the potential to falsify the most general Darwinian model. For instance, if it had been the case that eyes were not heritable then a Darwinian model would be rejected. A Darwinian model would also have been rejected if eyes did not increase the expected number of offspring of the bearers. The distribution of genes coding for eyes in modern taxa must also be consisted with common
ancestry and known ways of transmitting genes from one generation to the next. The best way to disprove Darwin is not to test the theory in
isolation, though, but rather to formulate a new model which predicts the data with higher accuracy and generality, but using fewer free parameters.

In conclusion, Berlinski's letter does not seem to significantly address any aspect of Nilsson & Pelger's work, but it does contain a few strange notions.
-------------------------
* Or population-of-genotypes space, maybe, which would be extremely general.
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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.
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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?
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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.
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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").]
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