Well, you’re right. In fact, the 9 out of 10 picture is when you don’t factor in the biological and genetic noises. When you do factor them in, then you can get the 99,9% picture. Here is the relevant passage in Sanford’s book:
When Mendel
So no point paying attention to anything else in that paragraph. Mendel’s Accountant is a garbage program hard-coded to give the results Sanford wants.
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That’s fine. I asked for a proof, though. The full thing. Not just premises, or, as you call them, “assumptions”.
That’s irrelevant. I didn’t ask you to convince me, I asked you to present a proof. Please, prove that the possibility of a search is contingent upon functional sequence space size. Proofs don’t begin with queries into agreement, nor with pleas to any. Direct proofs begin with the theorem’s antecedents, continue with inferences from said antecedents, and culminate in the theorem’s consequens. Indirect proofs begin with an assumption of the falsity of the consequens and derive by way of logical inferences a contradiction with the conjunction of the antecedents, lastly utilizing the equivalence of the contrapositive to conclude the validity of the theorem. Anyone who spent any time at all studying any maths at all knows this. But we have done pretty much everything at this point to establish that you lied about that, shy of subjecting you to a test, same how @Meerkat_SK5 met his final disgrace in this very thread not too long ago.
I voiced none.
Then there is nothing stopping you from showing said work. You choose not to. Perchance you prefer looking like a liar for no particular reason. Suit yourself.
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With realistic population sizes and so on Sanford’s program can’t run on any typical desktop computer. It absolutely cannot simulate the population genetics of a typical bacterial population in a small 100ml culturing flask, for example.
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This is not what has been discussed. I corrected you before that the probability of a search is based on the ratio of the functional sequence space to the total sequence space. In the beginning we were discussing skeletal alpha actin with a total sequence space of 20^371.
A definition of probability is:
the extent to which an event is likely to occur, measured by the ratio of the favorable cases to the whole number of cases possible.
Since functionality can be assigned as a favorable case is my proposition true by definition?
The probability of a search being successful is contingent on the size of the search. The size of a search is based on the above ratio.
What do you think is left to prove here?
Nothing. You have defined yourself as being correct. I have no comments to contribute to that. You can proceed with your questions now, if you wish.
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Let me correct that for you Bill:
I [asserted, without any evidence] before that the probability of a search is based on the ratio of the functional sequence space to the total sequence space.
This is a very bad definition of probability, as it contains the implicit assumption that all “cases” are equally probable.
You have flunked High School Statistics yet again.
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Everything I have said is correct as I have done the research and simplified the problem down to a definition that exists in the public domain.
You assigned shame to @Meerkat_SK5. Hopefully we’re here to learn from each other and not shame each other.
There are proofs available for building the equation n^l from the three basic probability axioms starting with proving P(a) and P(b) independently occurring is P(a) xP(b). If you feel you have special expertise in generating probability proofs I would be interested in discussing these. The ones that I have seen seem cumbersome.
As I said, the way you define probability renders your statement trivially, definitionally correct. I am of course free to point out that the definition you provided is grossly inadequate to handle the vast, vast majority of statistical problems, but that doesn’t change the fact that, if we were to restrict ourselves to the handful of problems it is adequate for, within that framework your statement would be correct. Hence, I have no comments to contribute to that. You can proceed with your questions now, if you wish.
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Ok, some common ground. Thanks
My question is do you have a simplified proof (using the three probability axioms) for the independent events P(a) and P(b) occurring being calculated as P(a) x P(b). Hopefully without using conditional probability.
This seems to be the foundational proof for the equation for N^L=TSS (total sequence space)
You have a habit of botching the terminology that makes no sense. What on earth is the “probability of a search” ? I wouldn’t call the probability of a single guess hitting the target, when all cases are equiprobable “the probability of a search”.
That is an awful definition. It will only work if you break the problem down to equiprobable cases.
A search is not a single guess. Nor need it be based on pure chance. And I certainly wouldn’t call that ratio “the size of a search”.
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You mean Kolmogorov’s axioms? No, I don’t. They do not define what it means for two random variables to be independent, so they are not sufficient to derive anything about them.
That being said, the most common definition of events being independent is one where the probability of one occurring on the condition that the other does is the same as without that condition. The inference, that therefore the joint probability (or probability of the intersect, as it were) equals the product of the probabilities follows by a rather brief (if not immediate/trivial) rearrangement, given the definition of the conditional probability.
Do you want me to write that up, formally, with the commonly used definitions and all? Again, the axioms don’t really come into play here. In fact, the way the quantities in question are defined, if we forget that they are supposed to be probabilities for a second, the axioms might as well not even need be assumed for this one in particular.
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Ok. The proofs I have seen require the axioms especially #2 P(S) = 1 where S=sample space.
When you have time I would very much like to see a write up. Thanks.
Everything you have said is wrong. You still don’t understand that
- How functional sequences are clustered
- The fraction of the space that is functional
- That evolution isn’t a blind random guess
are the relevant criteria. Not the total size of the space.
So Bill, what is
A) Your model for how the “search”(evolution, not random blind guessing) occurs?
B) Your estimate of how densely packed functions are?
C) Your numbers for the fraction of space being functional?
You need all three factors. N^L is useless by itself. It’s flat out useless.
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Sure, here goes. I’ll use your notation of S as our sample space.
Definition 1: The conditional probability, read as “Probability of A, given B” for two events A,B\subseteq S, where P(B)\neq0, is denoted by P(A|B) and defined thus: P(A|B):=\frac{P(A\cap B)}{P(B)}.
Definition 2: Two events A\subseteq S and B\subseteq S are called independent if and only if P(A|B)=P(A).
So, if A and B are independent, we know that P(A|B)=P(A) by Def. 2, therefore \frac{P(A\cap B)}{P(B)}=P(A) by Def 1. Multiply both sides of the latter by P(B) and we see: \frac{P(A\cap B)}{P(B)}\cdot P(B)=P(A\cap B)=P(A)\cdot P(B).
Side note: If you just google independence in the context of probability, among your first results will be the Wikipedia page. There, they define independence to mean that the probability of the intersect equals the product of probabilities. That’s fine, too. In that case my Def 2 would be a theorem that immediately follows from their definition of independence and Def 1 as here given.
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This is a totally unsubstantiated and as such null and void claim.
I’ve used the program and skimmed the code.
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Ok. Then let show us how the program is
hard-coded to give the results Sanford wants.
Sure, just open it up and look at the way selection is handled by the algorithm. That is… not handled.
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Is it even possible to run the program with settings where fitness increases?
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