Just to make sure AK reads it.
On Wednesday, January 18, 2012 2:13:33 PM UTC-5, Alan Kleinman MD PhD wrote:
> On Jan 17, 9:25 pm, hersheyh <
her...@yahoo.com> wrote:
> > On Tuesday, January 17, 2012 6:22:05 PM UTC-5, Alan Kleinman MD PhD wrote:
> >
> >
> >
> >
> >
> > > On Jan 16, 7:31 pm, Charles Brenner <
cbr...@berkeley.edu> wrote:
> > > > On Jan 16, 6:33 pm, hersheyh <
her...@yahoo.com> wrote:
> >
> > > > > On Friday, January 13, 2012 12:46:08 PM UTC-5, Alan Kleinman MD PhD wrote:
> >
> > > > > > On Jan 11, 3:07 am, Charles Brenner <
cbr...@berkeley.edu> wrote:
> > > > > > > On Jan 10, 1:28 pm, Alan Kleinman MD PhD <
kle...@sti.net> wrote:
> >
> > > > > > > > It is clear that no evolutionist posting on this thread understands
> > > > > > > > how to do the mathematics of mutation and selection
> >
> > > > > > > "the"? I do not think you understand the meaning of the definite
> > > > > > > article in English.
> >
> > > > > > > And it is also certain that you mean nothing by the word "selection."
> > > > > > > Whatever else may be said of your trivial probability comments alleged
> > > > > > > to be an answer, they in no way relate to selection. (I am not the
> > > > > > > first to note this.)
> >
> > > > > > > > We start with the following definition P(-∞ < X < +∞) = 1.
> >
> > > > > Can you please parse what you think you are saying here?
> > > > > If you are saying that the probability of an event X, when X
> > > > > is any value between minus and plus infinity, is always equal
> > > > > to one seems like a strange thing to say. The event of rolling
> > > > > a 6 with a die lies between minus and plus infinity, but the
> > > > > probability of rolling it is (for an honest die) about 1/6, not 1.
> >
> > > > The notation is ok. Realize that P(E) means that E is a statement
> > > > (which can be either true or false), and P(E) is the probability that
> > > > it is true. Alan has written -∞ < X < +∞ for his E. The event isn't X;
> > > > it's the whole statement inside the parentheses. We interpret the
> > > > statement E as meaning that X is a random variable (meaning a function
> > > > that takes on different values at different times), and X is claimed
> > > > to lie in the range from negative to positive infinite -- in other
> > > > words, E is the statement "X is a number on the real line". To say
> > > > that P(E)=1 then means saying that X is always on the real line, i.e.
> > > > it is just a fancy way to say "Let X be a real number".
> >
> > > Here are some simple examples which demonstrate this notation if X is
> > > the number that turns up in rolling a fair die, P(X = 1) = 1/6, P(X =
> > > 2) = 1/6 etc., P(1 < X < 2) = 0, P(1 ≤ X ≤ 2) = 1/3, P(0 ≤ X ≤ 3.2) =
> > > 1/2, P(X > 4) = 1/3, etc. And if we write P(-∞ < X < +∞), we are
> > > including all possible outcomes from a random trial.
> >
> > Apparently you are using the terms plus and minus infinity
> > metaphorically. What you really mean is:
> > P(all possible results) = 1. Or the sum of the probabilities of
> > every possible result = 1.
>
> I’m using the term both metaphorically and mathematically.
No. It only makes sense metaphorically (and not even that).
Mathematically it makes no sense to talk about the probability
of a number between -∞ and +∞.
> > > However, X is not a real number. X is the label we assign to the
> > > outcomes. If we took our six sided fair die and instead of labeling
> > > the sides with a 1, 2, 3, 4, 5, and 6 we used the labels Ad, Gu, Cy,
> > > Th, hersheyh, and Charles Brenner, then P(X = Ad) = 1/6, P(X = Gu) =
> > > 1/6 … P(X = hersheyh) = 1/6, P(X = Charles Brenner) = 1/6.
> >
> > How does P(-∞ < Kleinman < +∞) = 1 make any mathematical
> > sense at all? I certainly agree that Kleinman is not a real number
> > or any number at all. It is a name applied to an event or a set of
> > events with certain properties, just like "mutant" is a name applied
> > to an event or a set of events with certain properties.
>
> X is simply any possible outcome for a trial.
Even metaphorically, X would have to be *all* (not *any*) possible outcome[s]
for a trial. You do know that there is a difference between saying "any" and
saying "all", don't you? But then why wrap it between two infinities, instead
of just saying P(X), where X is all possible categories = 1?
> We can label those
> outcomes any way you want without affecting the mathematical behavior
> of the stochastic process. We could label the sides of dice with 10,
> 20, 30, 40, 50, and 60 but the probabilities for a particular side
> appearing remain the same.
My problem is that if you are going to be claiming something mathematical,
it should make mathematical sense. That is, what you claim you mean and
the mathematical representation you present should say the same thing
rather than different or nonsensical things.
> > If you want to say that the sum of the probabilities of all possible
> > results = 1, that would make sense. That is, if there are 6 possible
> > different results from tossing a die, the sum of their individual
> > probabilities would = 1. That is the addition rule of probability.
> > That would be true regardless of whether the individual probabilities
> > are equal to each other or not.
> And that is correct way to use the addition rule because now we are
> talking about mutually exclusive events.
I *know* what *I* said (and what you probably *meant*) is the correct
way to use the addition rule. But that isn't what P(-∞ < X < +∞) = 1
says. It says that the probability of *any* single *number*, X, between
-∞ and +∞ always = 1. That is not math. It is the arrogant pseudomath
of someone who thinks he can write an equation and make it mean
whatever he wants it to mean.
> > If I took a fair six-sided die, and labelled one side 1, two sides 2
> > and three sides 3, the probability of a 1 would be 1/6, the
> > probability of a 2 would be 2/6 = 1/3, and the probability of
> > a 3 would be 3/6 = 1/2. And the sum of the individual
> > probabilities would be 1/6 + 2/6 + 3/6 = 1.
> Again, that is the correct way to use the addition rule. And as I
> showed previously, if you weight the die so that you increase the
> probability that a particular face will show, you will decrease the
> probabilities of the other faces appearing.
I understand that. Which is also why, given that new mutations
are rare events, that *nearly* all the time, the face you see would
be the "not-mutant" face. That "not-mutant" face will have one
of the four bases, if you insist on defining mutation as an event
at a particular nt site. Absent past selection or past drift, the
probability of point or any other kind of "mutant(s)" at a nt site
in a population will be quite small relative to the probability of
not-mutants. That is quite unlike your expectation that the probability
of point mutation will be 1/4 with the presence of bases in equal proportion.
> > Just as the sum of the probability of a "mutant-of-interest" (one
> > event or set of events with certain properties) and "not-mutants"
> > (another event or set of events with different properties) = 1.
> > Note that, for example, if you define "mutant" as being the set
> > of possible point mutants, there would be three possible point
> > mutations (changes from the non-mutant base) in that set.
>
> Hersheyh, you continually try to force the mathematics of mutation and
> selection into the binomial model.
I am not "forcing" the math of mutation (selection has a different math)
into a binomial framework. That is the most natural and parsimonious
and elegant way of presenting the math of mutation. You do it by classifying
everything one sees as either having the properties you are interested in
(the mutation(s)-of-interest) or not having the properties you are interested
in (the not-mutation(s)-of-interest).
Very simple yes/no rule determined for each trial: Mutant-of-interest
or not-mutant-of-interest.
> Because of this, you are unable to
> do the accounting for all the different possible forms of mutation.
Nope. I can account for all possible forms of mutation by defining
what I consider a "mutation". For determining the probability of
'point mutation', for example, the original base in the population is
considered to be a "not-mutation-of-interest" and the three bases
that the "not-mutant-base" can change to are, collectively, the
"mutation(s)-of-interest". Mutations other than the change to the
three other bases are also included as "not-mutation-of-interest"
because they are not point mutations. Thus, you easily have two
categories: point mutations and not-point-mutations. To determine
the probability of point mutations in a population, you count the
number of individual organisms that have a point mutation and
divide it by the population size. You don't have to make any
stupid false assumptions about equal distributions. You don't
have to consider deletion mutations, because we know the
number of point mutations and the population size and know
that p(u) + p(~u) = 1. p(u) is the sum of the probabilities of
all the changes we have defined as being the "mutant(s) of
interest" and p(~u) is the probability of everything else.
If I define (as you seem to do) the mutation I am interested in
as "a change from the original base at a specific nt site to one
specific different base that causes a change in phenotype which
is called beneficial in a specified environment", then only that
specific change represents the "mutation-of-interest" and the
original base at that nt and all other kinds of mutation, including
point mutations other than one described as beneficial are
classified in the "not-mutation-of-interest" binary category.
If I define the "mutation-of-interest" as *any* mutation in a
specified *gene* (or codon or even nt) that makes the organism
better fit in a specified environment, then any mutation in that
gene that has that effect is classified as the "mutant(s)-of-interest"
and those that don't are classified as "not-mutant(s)-of-interest".
If I define the "mutation-of-interest" as *any* mutation that makes
the organism better fit (or just different) in a specified environment,
then any mutation anywhere in the genome that has that effect is classified
as a "mutant-of-interest" and all other mutations in that genome
are classified with the non-mutant sequences as "not-mutant-of-interest".
The probability of a mutant per trial, p, is the # of "mutant(s) of interest"
divided by the # of genes or nts or organisms examined for "mutant", as
clearly defined. This probability, p, is different than the mass probability
of one or more mutants in a population of n trials when the probability
of the mutant is p.
That probability, the probability of one or more mutants in a population
is 1 - (1-p)^n.
> > But if that is what you meant, you have a very strange way of saying it.
>
> It only sounds strange to you because you have never heard the
> mathematics on mutation and selection described properly before.
And anyone who thinks *you* are correctly describing it is a fool.
Your math of random mutation is incorrect in many places and
your math doesn't describe selection at all.
> You
> have gotten hung up on the first step of the derivation of the
> probability of a particular mutation occurring at a specific locus by
> the weight factor of 1/4 on the mutation rate when considering only
> point mutations in the mathematics. That weight factor will change
> depending on the particular mutation being considered.
Scientists are interested in the *real* probability of mutants (and new
mutation in particular) in a population, not an airy-fairy hypothetical
probability making false assumptions about the nature of mutation.
Your assumption that point mutations somehow does not involve
a *change* from the "not-mutant" base to one of three "mutant" bases
is one such false assumption. The assumption that mutation always
is described as a change at a nt site and that probability is representative
of all mutation is another.