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Author
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Topic: Co-Evolution
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YZ2
Member
Member # 91
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posted 11. December 2002 13:24
I have been thinking about estimating the likelihood for co-evolution of complex object. The following is my preliminary thought:
Let X=”ABCD” be the current state of the complex object with many attributes;
Y=”ABCDE” be the favored state of the fitness function when X=”ABCD”.
Want estimate:
P1= P(X=”ABCDE”, Y=”ABCDE”), joint probability that X evolves “ABCDE” given that the favored state of the fitness function is also Y=”ABCDE”
This is important for survival that the outcome of X is the same as Y, or fit in a particular case.
Assume P(Y) is conditioned on the current state of X=”ABCD”, (for co-evolution) then
P1=P(Y=”ABCDE”, X=”ABCDE” | X=”ABCD”)
or estimated as:
P1= P(Y=”ABCDE”|X=”ABCD”)*P(X=”ABCDE”|X=”ABCD”) ----------(1)
or roughly rewritten as:
P(next favored state of Y=”E”)*P(next state of X=”E”) given the current state of X=”ABCD”
Then: The rate of decreasing probability of P1 is in the order of O(P(X=”E”)) or with just one attribute of X
However, if P(Y) is not conditioned on (independent of) X=”ABCD”, then equation (1) becomes:
P1=P(X=”ABCDE”|X=”ABCD”)*P(Y=”ABCDE”)
Consider that the rate of decreasing probability of P(Y=”ABCDE”) is much faster than
P(X=”ABCDE”|X=”ABCD”), we drop this term and describe P1 of improbability in the order of O(P(Y=”ABCDE”)).
In other words, P1 is as small as P(Y=”ABCDE”), or the order of improbability is: O(P1)=O(P(Y=”ABCDE”))
This means that there is no gain in likelihood in co-evolution if the favored next state of the fitness function is independent of the current state of the complex object.
Now if there is really no knowledge about whether P(Y) is conditioned on X or not on average (despite the fact that sometimes, it is conditioned on X), whether it is Y=”ABCDE” or any other states, it is then as likely or unlikely as P(X), or similar to a blind search in the worst case (or a displacement scenario).
That is, if P(Y) conditioned on X cannot be determined in general, their co-evolution has to be determined on a case-by-case basis, similar to evaluating P(X).
[Moderator Note: I capitalized the title of the post]
[ 11. December 2002, 13:48: Message edited by: Moderator ]
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YZ2
Member
Member # 91
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posted 11. December 2002 14:58
One more point:
If P(Y) is conditioned on X, then it is analogous to the existence of a global fittness function that fits the whole sequence of X, rather than a local one.
[ 13. June 2003, 10:40: Message edited by: YZ2 ]
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RBH
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Member # 380
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posted 11. December 2002 15:47
In order to understand David Chiu's posting I have a few questions of clarification to help me understand the post. Chiu wrote quote:
Let X="ABCD" be the current state of the complex object with many attributes;
Y="ABCDE" be the favored state of the fitness function when X="ABCD".
Want estimate:
P1= P(X="ABCDE", Y="ABCDE"), joint probability that X evolves "ABCDE" given that the favored state of the fitness function is also Y="ABCDE"
This is important for survival that the outcome of X is the same as Y, or fit in a particular case.
Questions:
1. Is what is being considered a single entity or is it a population having members with those attributes? What is the unit of analysis for which probabilities are to be estimated?
2. Are "A", "B", ... "E" variables that take values - (i.e., are they dimensions of a fitness landscape?) - or are they attributes with fixed values?
3. In "Y=ABCDE," is "Y" some other point (different from the point X=ABCD) on the fitness landscape(s) on which X is currently situated, or does the addition of "E" mean that a new attribute is added to the fitness function that therefore induces new and higher-dimensioned fitness landscape(s) for X?
4. What does "favored state" mean? Is the current state (X=ABCD) also "favored" in the sense of having some relatively high fitness value?
Chiu further wrote quote:
In other words, P1 is as small as P(Y="ABCDE"), or the order of improbability is: O(P1)=O(P(Y="ABCDE"))
This means that there is no gain in likelihood in co-evolution if the favored next state of the fitness function is independent of the current state of the complex object.
I interpret this to be a statement about a statistical property of the topography of the fitness landscape(s) on which the entity (or population) is situated. If that interpretation is correct, then the "if" conditional could be reworded "...if the topography of the fitness landscape(s) is uncorrelated." Is that the case?
Finally, in his addendum Chiu wrote quote:
If P(Y) is conditioned on X, then it is analogous to the existence of a global fittness function that fits the whole sequence of X, rather than a local one.
Should one interpret that to mean that different 'regions' of a given fitness landscape do not differ in their topographical correlation and that a given fitness landscape has a statistically homogenous (at least with respect to correlation) topography?
Thanks!
RBH
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YZ2
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Member # 91
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posted 11. December 2002 17:14
I am not sure I fully understand your questions. I’ll try to see if these comments are helpful.
In my formulation, X and Y are different variables. The outcome of X is a complex object with multiple attributes that I denote as “ABCD” etc. “A” “B” and so on are the individual attribute. X=”ABCD” thus denotes an instance of an object with attributes “ABCD”. P(X=”ABCD”) denotes the a priori probability of X with the described attributes.
From X=”ABCD” to X=”ABCDE” denotes the object with attributes “ABCD” acquires additional attribute “E” as “ABCDE”. It describes an occurrence of the attributes for the object.
The outcome of Y is the favored state of a (current) fitness function (that may change and evolve). It is not a topography, but only describes that point of fit (as a simplification). So Y=”ABCDE” denotes that it favors objects with attributes “ABCDE”.
I think the argument can be generalized to topography, but as a point is simpler.
Hence the joint occurrence of (X=”ABCDE”, Y=”ABCDE”) denotes that object X=”ABCDE” is also favored by the (current) fitness function. For example, other joint occurrences, say (X=”ACFDE”, Y=”ABCDE”) are objects not favored and may lead to death of the object.
I use the big-O notation to calculate the (decreasing) growth rate of the probability estimate when more attributes are acquired for comparison.
[ 13. June 2003, 10:40: Message edited by: YZ2 ]
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RBH
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posted 12. December 2002 00:41
Hm. OK, I think I have a bit more of a handle on what you're saying. For now I believe I'll lay back and see how this develops, particularly how it develops as a model of co-evolution. I'm not seeing that connection yet.
RBH
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YZ2
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Member # 91
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posted 18. December 2002 11:33
Actually, what I have been wrestling with here is not just a model of co-evolution. I have been thinking about a unifying model for co-evolution and design. Let me explain. I have defined X as the occurrences of a complex object and Y as the fitness function for X. Their independence/dependence cannot be (easily) determined in general, even though evidence may be available for specific cases. If the fitness function Y is conditioned on X, then their properties can be understood by the evolving process of X. This is the view based on co-evolution. If Y is independent of X, then the occurrence of X has no immediate bearing on Y and vice versa. However, assuming an equilibrium is achieved, Y can be estimated by E(X) where E is analogous to the “expected value of X”. This is not the same as the traditional expected value since X are complex objects. To understand Y more in this case, we can eliminate “chance”, “noise”, “variations” and the like of X in calculating E(X). (This is not obvious and can be an important area of research.) What remains after elimination estimate the nature of Y. In fact, many of the proposed analysis of “design” can then be applied (together with their various degree of success).
As a scientific paradigm, in cases when Y is known to be conditioned on X, the “evolutionary process” should be applied to understand X and Y. However, when Y is independent of X, then “design” analysis is more appropriate. Therefore, a reasonable approach is to evaluate the likelihood of dependence/independence between X and Y in a particular case before performing the appropriate analysis. Alternatively, in cases when their dependence/independence cannot be determined, both analyses can be performed and their fruitfulness evaluated.
What is intriguing here is that it is a unifying theory that “evolutionary process” and “design” are special cases. If independence/dependence between the complex object and the fitness function cannot be asserted absolutely, then either theory alone is inadequate. Does this unifying theory provide more than both theories combined? I think the answer is yes. This is the case when Y is only conditioned on X partially in some outcomes but not the others. In this case, using both analyses may generate more comprehensive results.
I plan to take some time off for the holiday season. I sincerely wish all the brainstomers a merry Christmas and a very happy new year. I am sure the moderator can do a very good job to couch this post along. Please do not hesitate to continue the discussion. Many of the heated and cool thoughts will certainly be as good as mine.
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