Felsenstein's Cost Paradox
- From: "john edser" <edser@xxxxxxxxxxxxxx>
- Date: Thu, 6 Jul 2006 13:52:48 -0400 (EDT)
JE:-
Felsenstein has (predictably) refused to answer any of the questions that I
put to him in my last post with regards to this apparent cost paradox which
he freely identified within Haldane's Dilemma. Please refer to the end of
this post for the historical details as to why he has always refused to
reply.
Here is a repost of this paradox as it was outlined by Felsenstein:
---------------------------------- start of repost
----------------------------
Joe Felsenstein <joe@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
This is quite interesting and leads to the following puzzle. Supposeconstant.
there are two alleles, A1 and A2 and the population size remains
They are in equal frequencies at the start. In time period 1, as aresult
of natural selection, A1 increases to 0.52. There is a modest cost(Remine's
cost) for this. Then A1 decreases in the next period to 0, under adifferent
selection regime. No extra cost. The total cost has been small.looked at
But ... suppose we ask about the other allele, A2. In the first time
period it decreases to 0.48 (at no cost). Then in the next time period
it increases to 1.0, with lots of cost involved.
But these are the same events. Looked at one way the cost is small,
the other way it is large. Which is true?add the
Perhaps one is supposed to
cost for increases of both alleles, in which case why are we told thatthe cost
of a decrease of one allele is zero when, if there is only one otherallele,
there is necessarily a cost for its resulting increase?
JE:-
A solution requires an unambiguous definition and complete separation of two
quite different cost structures:
1) The absolute (total) cost concept which cannot be zero.
2) Just the relative comparative cost concept which can be zero.
Both absolute and relative cost structures always operate but not
simultaneously (one must precede the other).
Here are the required definitions:
1. Absolute cost: The total cost of any defined _independent_ event.
2. Relative cost: The SUB total costs of just RELATIVE _dependent_ events
defined to constitute the ONE SAME independent event.
Where:
In time units, the absolute cost must always precede a relative zero cost
unless you loop them tautologically allowing time to go backwards. As
amazing as this may seem, this type of mistake is easy to make within an
oversimplified mathematical model because no constant term exists to provide
the (missing) critical frame a reference.
In Felsenstein's Cost Paradox the minimum absolute non comparative cost
which cannot be zero and which has to always be paid is just 1. This is
because both alleles remain empirically fertile organism _fitness dependent_
and not just heuristically gene centric _fitness independent_. The minimum
absolute cost of substitution is simply the size of the defined constant
population in fertile organism units. Substituting alleles within just an
infertile population fails as does the argument that it is much cheaper to
substitute alleles within immature infertile forms. Unless they are raised
to fertile adulthood they remain useless vehicles of substitution.
DETAILS:
This ABSOLUTE cost is calculated as follows: Felsenstein's "modest cost" of
increasing the A1 allele from 0 to 0.52 in time unit 1 is added to A2 where
it _dependently and not independently_, decreased from to 1 to 0.48 "at no
cost" also in time unit 1.
0.48+0.52 = 1
Only the RELATIVE costs in each case MAY be different for each allele. These
were 0.52 for A1 and 0.48 for A2 where by logical necessity, 1-0.48 = 0.52.
On just a relative basis A2 paid relatively less for the gene freq changes
produced from time unit 1 to time unit 2. Note that the largest possible
relative cost of substitution cannot exceed the maximal absolute total cost
unless "a free lunch" is provided but it can be smaller. The limit is set by
the absolute assumption in this case which is the constant population size.
I don't think that any of the above is unusual. IMHO all of it is just
common sense, if and only if, the Post Modern proposition that "everything
is relative" becomes substituted with "everything is based on absolute
_assumptions_" (which is the sciences must be refutable). As I mentioned
above the critical absolute assumption in this particular case is the
constant population size. Please note that no group selection is proposed
using this population constant because the population units remain fitness
interdependent and not fitness dependent.
Felsenstein's comments are diagnostic of the biological error which allowed
the paradox. The supposed "modest cost" of increasing allele A1 from 0 to 52
and Felsenstein's "no cost" of decreasing allele A2 from 1 to 0.48 simply
deleted their empirical fitness DEPENDENT association creating just the
mathematical _illusion_ that they remained independent costs. In this purely
hypothetical situation no frame of reference actually existed allowing the
cost paradox which he identified.
Answering my own question: Remine's cost is a result of the increase of
numbers, not frequency of an allele. So if allele A1 decreases in numbers
but A2 remains the same (hence the population size decreases), there is no
cost. But if population size remains constant as A1 decreases, then A2
necessarily increases and there is a cost from that.
JE:-
Felsenstein's solution employed to only referring to increasing "numbers".
Unfortunately this represents a nonsensical biology because numbers HAVE TO
represent a biological entity. His use of the mathematical term "number"
appears to be an attempt to evade the inevitable paradox created by his
continued refusal to allow a testable frame of reference via a defined
constant term. This is required to remove the apparent paradox.
I ask for a SECOND TIME: would Prof. Felsenstein please unambiguously state
what these "numbers" represent within the biological sciences ?
FOOTNOTE:
Felsenstein's continued refusal to reply to any of my posts is based on my
(understandably unpalatable) accusation that the fitness accounting scheme
which he continues to employ within evolutionary theory (I include Haldane's
Dilemma) contains the same epistemological error as the Enron accounting
scheme within which debits were incorrectly allowed to be written up as
credits. The Enron scheme only became possible because a critical frame of
reference was removed, i.e. it was incorrectly allowed as just a misused
accounting oversimplification.
Historically, I requested Felsenstein to provide a comment re: the quote
below which concerns a similar problem re: the deletion of a critical frame
of reference from within Hamilton's oversimplified rule:-
--------------quote----------------------
1) 22/01/2004:
JE:-
What is the difference between
a reduced positive c and a negative c?
If c was an absolute measure of fitness
then yes, a real difference exists. However
c is only a relative fitness cost and not
an absolute fitness cost, so what is the
difference?
BOH:-
As far as the rule is concerned, none.
----------- end quote --------------------
"BOH" who provided the illuminating response within this reposted quote is
Dr Bob O'Hara who posted here as a professional in this field.
The following is the last posting of the thread "Hamilton's rule: the
high cost of a free lunch" from 13th of January 2004 -
------------------- start of quote -------
JE:
Any "involvement" by yourself in the Enron
scam appears to be epistemological, and by
default. Because you refuse to
just discuss cause and affect denying
any accountability of it actually being
reversed within a false accounting scheme,
then by simple default, you have sanctioned
the action of these accountants.
JF: [meaning Joe Felsenstein]
That is truly offensive. It is not humor but
ill-tempered accusation. Apologize!
---------------- end of quote ----------------
It should now be clear that I refused to apologize for good reason. Allowing
the deletion of a critical theoretical constant can sanction the reversal of
empirical cause and effect within ANY oversimplified model because no frame
of reference exists. Felsenstein's Cost Paradox repeated the same error I
was referring to then: the deletion of a critical frame of reference from
within Hamilton's Rule which constituted the same type of error as the Enron
accounting scheme deletion. This type of deletion was simply repeated within
the cost structure of Haldane's Dilemma providing the cost paradox which
Felsenstein freely identified.
All these ongoing modeling deletions constitute examples of oversimplified
model misuse. Each model was oversimplified from a theory, which alone,
provided some critical frame of reference. The constant term which allowed
this within each theory was deliberately deleted from within the proposed
model of that theory. This model then went on to illegally contest and
finally replace that parent theory (the theory the model was oversimplified
from).
Regards,
John Edser
Independent Researcher
edser@xxxxxxxxxx
.
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