While I am aware that @Paul_King has provided the requested proof, the fact that the proof does not require “establishing P(a) x P(b) or the independent probability of a and b occurring using only the 3 probability axioms” should be obvious from the fact that the “the N^L (number of units per set and length of the sequence) calculation” is firmly in the realm of Algebra, and whilst Probability theory requires Algebra, Algebra does not require Probability theory.
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My pure maths instinct is to start at L=0, not L=1. Step 1 is simpler then.
My pure maths gremlin wants to try L=-1…
I’ve never encountered these before - are they a recent tesching aid? They’re not really axioms.
And why isn’t the first axiom 0 <= p <= 1 ?
Hi Dan
This is what I asked for. Do you have any comment on the inter relationship between combinatorics and probability?
These probably aren’t the same as the ones in my Casella-Berger textbook, but I didn’t want to dig that out.
There are generally more than one equivalent way to state basic axioms. Here you can get 0<=p<=1 from the other 3 given in that link.
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Other than "you don’t seem to be using either,* no.
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And considering that the meaning of independence in this context either is that the probability of the intersect is the product of the probabilities, or is defined in terms of conditional probability, no: There is no proof of that, whether it is a definition or a trivial rearrangement of definitions, that either involves Kolmogorov’s axioms or does not involve conditional probabilities. You will not meet someone who simultaneously both knows the subject and has what you asked for.
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The only relationship is a trivial one. If all the possible outcomes are equiprobable then the probability of each is the inverse of the number of outcomes. Combinatorics is a way to count outcomes (eg the results of rolling two dice) but that’s it.
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Combinatorics is necessary for (some branches of) probability. Probability is not necessary for combinatorics. It is thus a purely one way relationship.
This seems reasonable however in this discussion we are using combinatorics to help build a probability calculation. In this case does it make sense that we hold to the basic axioms of probability in our calculations?
Are we? In order to do so, we’d first have to establish that “outcomes are equiprobable”, or some other probability distribution. This is where issues such as clustering come into play.
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No. You are the one trying to do that. Because you think that the amount of elements in a finite sized sample space gives one a clue about probabilities of events from that space. Because that’s how you define what probability is. Noone else is trying to do this, because most of us know better.
The basic axioms of probability have so far not been utilized for any argument in the current discussion. They have only been mentioned, because you brought them up, but they have not explicitly come into play yet. Frankly, I’m not convinced that you’d even know when to use them, if you could. This might as well be Meerkat throwing in “uncertainty principle”, because it’s a technical term, but without anything like enough of a grasp of its meaning to actually formulate an argument around it.
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You don’t seem to be even trying to build a probability calculation relevant to evolution. To do that you would have to start identifying the relevant outcomes and their probabilities.
And really why would you ask that question? If you are working with probability theory you must follow the basic axioms of that theory. Nobody has suggested otherwise (unless you have)
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I suspect Bill is attempting non-Euclidean probability.
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The original discussion was based on a probability calculation with the numerator and denominator built with combinatorics. I think we agree at this point.
I still have interest in discussing the inter dependence of combinatorics and probability if anyone is interested. I see there may be more here than meets the eye.
Maybe they are coached by Ted Lasso?
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Your original assertion about “a probability calculation with the numerator and denominator built with combinatorics” was based upon the absurd, unsubstantiated and unrealistic assumption that all outcomes are equiprobable.
I think it is highly unlikely that you will find anybody interested in a discussion based upon this absurd, unsubstantiated and unrealistic assumption, other than to debunk that assumption.
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A calculation which appears to be irrelevant to the supposed point.
Although, come to think of it, while the denominator might be (but wasn’t) calculated with combinatorics I don’t see how you could calculate the numerator like that - at least not with present knowledge.
Unless you can give some reason to think that there is something of importance there isn’t anything to discuss. I think you’re just unwilling to admit that you jumped to an obviously wrong conclusion and exposed your ignorance of the subject.
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Nonsense distribution ? It is the one on which Kimura and most of population geneticists agree on, though.
Yes. But note that in his analyses using Mendel’s Accountant, he often sets this parameter at 1 in 1000 mutations, a choice purposely made to favor the evolutionary scenario.
The graph is adapted from a figure by Kimura, who, I guess, used data to generate it.
Correct
Again, as such, this is an unsubstantiated claim.
Of course. See fig 2 to 9 of the paper below, especially fig 8 and 9.
Why on earth are you claiming that?
Using MA, Sanford and colleagues were able to
- show that neutral mutations go to fixation just as predicted by conventional theory (i.e., over deep time the fixation rate approached the gametic mutation rate)
- show that « Haldane ´s dilemma » is a real thing
- correctly model Fisher’s fundamental theorem of natural selection
Ref: - the paper above
- "Haldane’s Ratchet" by Christopher L. Rupe and John C. Sanford
Not so bad, isn’t it?
With all due respect, that is not how I remember the sequence of events here. You assuming that probability is a matter of combinatorics does not make the discussion based on it. If memory serves, I challenged you explicitly to provide a proof that the “possibility of a search” is in some way contingent upon the sample space size, and after trying for some reason to instead substantiate the number of elements in a space that is overtly not the one under consideration, your final reply was to say that that counting measures is what you define probability in terms of; a definition that just about works for fair dice, but completely falls apart the moment nigh-on anything else is modeled. Even when, as you assume is mostly or always the case, atomic events are pairwise independent, let alone when talking of stochastic processes like what’s relevant in genetics.