Talk:Fisher's fundamental theorem of natural selection

message: There is a well-defined use of the word theorem in mathematics, which is linked to in the article. 
Describing something as a Theorem carries a strong implication of certainty, which is inappropriate for an idea which is either a statement of faith, a definition, or a first draft of a hypothesis, and where the terms used are so poorly defined that they have caused decades of controversy.
Using the word may have been a shrewd move by Fisher in the war for recognition. An encyclopedia should perhaps be a little distanced from such a claim.  — Preceding unsigned comment added by Fentlehan (talk • contribs) 09:46, 7 March 2012 (UTC)
replies: text: Fisher's original statement was a theorem.  It is not usually stated as such, but Fisher proved it as a theorem.  The article should reflect this, rather than discounting the idea as not a theorem (i.e. referring to it as a "so-called theorem").Trashbird1240 (talk) 19:27, 26 March 2012 (UTC)
thread_title:  Should the entry mark the fact that ''Theorem'' is a misnomer ? 

message: In the list of references, Edwards' 2000 paper in Genetics is listed. This paper is a historical review with anecdotes of things that have happened in the population genetics community, and only shortly mentions the fundamental theorem. I think it is out of scope for this wikipedia entry, so it could be deleted because of lack of relevance. Anyone who thinks otherwise? --Anthony Liekens 00:46, 26 Nov 2004 (UTC)
thread_title:  Edwards' genetics paper out of scope 

message: Why is the Edward's quote in the introduction?  It is not really more modern terminology, and there is confusion about gene frequency (should be allele frequency) and genic variation.  What does the Edwards' quote give us? Ted 15:47, 31 May 2006 (UTC)
replies: text: If it is a quote you can't change it to anything else than what Fisher wrote. You can find another quote or you can add information about the different types of genetic variance but don't change the wording of that quote.Sjö (talk) 04:52, 25 April 2012 (UTC)
thread_title:  Quote 

message: I have now been able to get a closer look on the proof of Fisher’s fundamental theorem.
Following Maynard Smith (see reference page 117), we may let the frequency of genotypes 
   g = (g1, g2, …, gn) before selection be 
   p = (p1, p2, …, pn) 
And their fitness 
   w = (w1, w2, …, wn).
Then the mean fitness, W, becomes (summation is over the set of indices i)
   W = Σ piwi);         (1)
After selection has operated, the frequency_after_selection of gi becomes
   pi* = piwi/W;               (2)               
and hence the mean fitness of of the selected parents W* is
   W* = Σ piwi2/W;
Hence the selection differential on fitness is
   S = W* – W = variance in fitness before selection / mean_fitness
Equation (2) is perfect as long as selection takes place on the level of individuals where w describes the fitness of the individuals. But Maynard Smith defines fitness as a property, not of an individual, but of a class of individuals - - (page 38). Suppose for instance that selection is triangular in a phenotypic 2-dimensional space (x,y) according to the figure below, and that the digits in x and y represent genotypes. The probability of selection for green points may be = 1 and for red points = 0. But this seems impossible if individuals have no fitness.
 
http://www.evolution-in-a-nutshell.se/select_triangle.gif
In order to secure the correctness, the fitness of genotypes must always - in accordance to equation (2) - be proportional to
   wi = W * frequency_after_selection / pi.       (3) 
Philosophically and numerically this seems tricky because W requires pi  and wi given beforehand according to (1), while wi requires frequency_after_selection and W given according to (3). I don’t know how to put it in English; Fisher’s theorem is formally correct, but the proof seems to go in a circle. What happens in the phenotypic space is not correctly dealt with.
On the other hand, Gaussian adaptation (GA) – using selection of individuals - gives the gradient (the steepest direction)  of W with respect to changes in m (the centre of gravity of Gaussian distributed phenotypes). The variance or variability of interest here is represented by M; the moment matrix of the Gaussian.
  gradient_of_mean_fitness = dW/dm = W M-1(m* – m)
where M is the moment matrix of the Gaussian and m* is the centre of gravity of phenotypes after selection. A possible correspondence to Fisher’s increase in W is
   S = (dW/dx)*Δx + (dW/dy)Δy
     = W(m* - m)’M-1(m* – m)
In this case the increase in W is from the offspring in one generation to the offspring in the next. It is assumed that the mutation rate is adapted such that M is fairly constant from generation to generation.
Simulations gave the results:
    Selection_differential = 0.0116      using the GA-gradient at black m
                           = 0.0112      by moving the Gaussian to dark-green m_aft
                           = 0.1295      According to Fisher.
It seems to me that the GA-model gives about the same result if the gradient is used for estimation, or if the Gaussian is moved to m*. But the small figures are uncertain and mean results of a hundred simulations have been used.
Thus, Fishers model gives a seemingly fast increase in W because the mutation rate and the offspring in the next generation is not considered. GA shows a very slow increase in this case because the triangle is thin. But both evolution and GA may adapt M to become more proportional to M*, which may speed up the process considerably. Therefore, I think that GA may contribute (a little) to the theory of evolution.
When GA is used for optimization the gradient will suffice.
Reference: Maynard Smith, J. Evolutionary Genetics. Oxford University Press, 1998.
--Kjells 09:11, 29 August 2007 (UTC)
replies: text: Maynard Smith presents a simplified version of Fisher's theorem, and his definition of fitness is not the same as Fisher's, so I am not sure your criticisms are relevant.  For Fisher the relevant concept of fitness is the relative rate of increase or decrease of the groups of individuals bearing particular gene variants (alleles).  I don't think it is now disputed that his theorem is mathematically valid, given his definitions.  It remains disputed whether it is as important as he thought. He certainly never claimed that there would be a rapid increase in fitness, or indeed that *absolute* fitness (measured simply by the number of surviving offspring), would necessarily  increase at all.  He recognized explicitly that the 'environment' - including the biotic environment - might be deteriorating at a rate sufficient to offset the increase in relative genetic fitness.86.183.203.119 (talk) 18:14, 18 September 2014 (UTC)
thread_title:  A closer look at Fisher's theorem 

message: A discussion about Fisher's fundamental theorem has recently been held at ScienceBlog, where I have been encouraged to publish some new paper about it. Here is my blog:
Submitted by kjellstrom on Sat, 2008-01-12 03:04. bioscience and medicine
 
Creationists have reason to doubt the theory based on Fisher’s fundamental theorem of natural selection published in 1930. It relies on the assumption that a gene (allele) may have a fitness of its own being a unit of selection. Historically this way of thinking has also influenced our view of egoism as the most important force in evolution; see for instance Hamilton about kin selection, 1963, or Dawkins about the selfish gene, 1976 in http://en.wikipedia.org/wiki/Gaussian_adaptation#References
On the other hand, if the selection of individuals rules the enrichment of genes, then
Gaussian adaptation will perhaps give a more reliable view of evolution (see the blog “Gaussian adaptation as a model of evolution”). 
In modern terminology (see Wikipedia) Fisher’s theorem has been stated as:
“The rate of increase in the mean fitness of any organism at any time ascribable to natural selection acting through changes in gene frequencies is exactly equal to its genic variance in fitness at that time”. (A.W.F. Edwards, 1994).
A proof as given by Maynard Smith, 1998, shows the theorem to be formally correct. Its formal validity may even be extended to the mean fitness and variance of individual fitness or the fitness of digits in real numbers representing the quantitative traits.
But, if the selection of individuals rules the enrichment of genes, I am afraid there might be a risk that the theory becomes nonsense, and that this is not very well known among biologists. 
A drawback is that it does not tell us the increase in mean fitness (see my blog “The definition of fitness of a DNA- or signal message”) from the offspring in one generation to the offspring in the next (which would be expected), but only from offspring to parents in the same generation. Another drawback is that the variance is a genic variance in fitness and not a variance in phenotypes. Therefore, the structure of a phenotypic landscape – which is of considerable importance to a possible increase in mean fitness - can’t be considered. So, it can’t tell us anything about what happens in phenotypic space.
The image shows two different cases (upper and lower) of individual selection, where the green points with fitness = 1 - between the two lines - will be selected, while the red points outside with fitness = 0 will not. The centre of gravity, m, of the offspring is heavy black and ditto of the parents and offspring in the new generation, m* (according to the Hardy-Weinberg law), is heavy red.
Because the fraction of green feasible points is the same in both cases, Fisher’s theorem states that the increase in mean fitness is equal in both upper and lower case. But the phenotypic variance (not considered by Fisher) in the horizontal direction is larger in the lower case, causing m* to considerably move away from the point of intersection of the lines. Thus, if the lines are pushed towards each other (due to arms races between different species), the risk of getting stuck decreases. This represents a considerable increase in mean fitness (assuming phenotypic variances almost constant). Because this gives room for more phenotypic disorder/entropy/diversity, we may expect diversity to increase according to the entropy law, provided that the mutation is sufficiently high.
So, Fisher’s theorem, the Hardy-Weinberg law or the entropy law does not prove that evolution maximizes mean fitness. On the other hand, Gaussian adaptation obeying the Hardy-Weinberg and entropy laws may perhaps serve as a complement to the classical theory, because it states that evolution may maximize two important collective parameters, namely mean fitness and diversity in parallel (at least with respect to all Gaussian distributed quantitative traits). This may hopefully show that egoism is not the only important force driving evolution, because any trait beneficial to the collective may evolve by natural selection of individuals.
Gkm
http://www.scienceblog.com/cms/creationists-have-reason-doubt-classical-theory-evolution-15214.html
http://www.scienceblog.com/cms/blog/kjellstrom
--Kjells (talk) 13:00, 16 January 2008 (UTC)
replies: text: Please don't add links to your own texts. As I understand it, talk pages aren't indexed by search engines, so the link here and on Talk:Gaussian adaptation won't increase the rating anyway.Sjö (talk) 17:24, 15 January 2008 (UTC)
replies: text: Thanks for the tip. --Kjells (talk) 13:00, 16 January 2008 (UTC)
thread_title:  Creationists have reason to doubt the classical theoy of evolution 

message: During discussions, I recently discovered something that is perhaps "well known", but is not covered here:  it would seem that Fisher unintentionally conflated two ideas: "variance of the fitness of a population" with the "genetic diversity of the population".  Based on practical experience with machine learning, its clear that these two concepts are distinct.  I've noticed that the rate of improvement in fitness is related to the genetic diversity of the population, and NOT to the variance of the fitness of the population.  That is, one can have algos that (unintentionally) generate a very homogeneous population, that has a large variation in fitness.  In such a situation, the algo grinds to a halt, and rate at which fitness improves becomes very slow.  The rate of improvement can be improved by increasing the diversity of the population!!
This, in a certain sense, I discovered that "Fisher was wrong", and that the correct statement should be: "The rate at which a population improves in fitness is proportional to the variance of the genetic diversity of the population".  (and that one should not confused "variance of the fitness" with "diversity", which are naively similar but in fact are quite unrelated).
Surely, I am not the first to observe this: but who is, and how can the article be amended to reflect this?   (p.s. this seems related to the phenomenon discussed in the post immediately above.) 67.198.37.16 (talk) 20:19, 20 April 2016 (UTC)
thread_title:  Genetic diversity? 

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