Hardy-Weinberg law



Guy Hoelzer wrote:
> Bob,
>
> in article [email protected], Anon. at
> [email protected] wrote on 6/23/04
> 9:06 AM:
>
>
>>Perplexed in Peoria wrote:
>>
>>>>In essence, any finite population will become inbred
>>>>over time (at least to some extent), and this increases
>>>>homozygosity.
>>>
>>>
>>>Boy, are you going to feel foolish after you get a good
>>>night's sleep and review what you have written :)
>>>
>>
>>I'm not sure I feel foolish, but I agree it would be
>>better to have written this:
>>
>>In essence, any finite population will become inbred over
>>time (at least to some extent), and this means an increase
>>in homozygosity.
>
>
> You are right that inbreeding is ALWAYS happening, because
> the individuals in every mating pair are ALWAYS related.
> This process causes there to ALWAYS be a tendency toward
> increasing degrees of both inbreeding and homozygosity
> within finite (all real) populations. Population
> subdivision and isolation by distance influence both of
> these effects, causing the degrees of inbreeding and
> homozygosity to increase at an even faster rate. These
> processes conspire to drive populations on the notorious
> "march toward homozygosity;" however, it is glaringly
> clear from empirical observations that this is a highly
> unbalanced view of nature. The data clearly demonstrate
> that outbreeding and mutation, which dynamically drive
> populations toward lower degrees of inbreeding and
> homozygosity, are usually sufficiently strong to stop the
> march well short of its endpoint.
>
You are, of course, right. I wasn't being clear that I was
talking about the models - that's where this all started
from. The models that we were discussing exclude population
structure and mutation, of course.

It looks like we're converging to agreement, so I might
avoid replying to a few threads if that's all we're doing.

Bob

--
Bob O'Hara

Dept. of Mathematics and Statistics
P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of
Helsinki Finland Telephone: +358-9-191 23743 Mobile:
+358 50 599 0540 Fax: +358-9-191 22 779 WWW:
http://www.RNI.Helsinki.FI/~boh/ Journal of Negative
Results - EEB: http://www.jnr-eeb.org
 
Tim Tyler wrote:
> Anon. <[email protected]> wrote
> or quoted:
>
>>>>>It's like claiming that half the integers are even.
>>>>
>>>>Err, they are. There are just rather a lot of them.
>>>
>>>No, there aren't.
>>>
>>>There are an infinite number of even numbers.
>>>
>>>There are an infinite number of odd numbers.
>>>
>>>Divide infinity by infinity and the result is
>>>indeterminate.
>>
>>If there are an equal number of even and odd numbers, then
>>half of the numbers must be even.
>
>
> This is not true when the sizes of the sets involved are
> infinite.
>
But they're the same size! We can count them!

>>This must be true because for every even number, I can add
>>1 and get an odd number. Conversely for every odd number I
>>can add 1 and get an even number. Hence, by the operation
>>of adding 1, I can produce an even number for every odd
>>number and vice versa. Ergo, half of all numbers are even,
>>and half are odd.
>
>
> I can easily create a map between every even number an 5
> unique odd numbers - i.e I can map from 2x to 5x+1, 5x+3,
> 5x+5, 5x*7 and 5x+9.
>
Yes, but that's not the operation of adding 1 is it?

What I missed out was a uniqueness statement (i.e. that I
can create all odd numbers uniquely from the even numbers).

> That is exactly the same sort of argument as the one you
> gave - yet it indicates that there are *five* odd numbers
> for every even number.
>
> What conclusion should one draw from this?
>
> The correct conclusion is that the argument you gave is
> useless.
>
Not quite, because I can resurrect it. For each integer (for
which there are a countably finite number), I can create a
single unique even number (i.e. I multiply the integer by
2). I can also create a single unique odd number: multiply
by 2 and add 1. There are therefore as many even numbers as
integers, and as many odd numbers as integers, so half of
the integers must be even.

>>>>>No serious mathematician can talk about fractions of
>>>>>infinite sets and expect to be taken seriously.
>>>>
>>>>But they do.
>>>
>>>No - not unless the fractions are "zero" or "one".
>>
>>Rubbish, unless you're denying the existence of fractions.
>>Fractions are fractions of an infinite set, because there
>>is an infinite number of numbers between 0 and 1 (proof:
>>take the reciprocal of every positive integer).
>
>
> 1/3 is *not* the ratio of the size of the set of numbers
> smaller than 1/3 and the size of the set of numbers
> greater than 1/3. That ratio is the ratio of two
> infinite numbers - and thus is not well defined.
>
So you claim, but have yet to provide a proof. This is
mathematics, so you should be able to provide a proof of
your assertion that one can never determine the value of the
ratio infinity/infinity.

>
>>>>It's how probability is defined as a concept.
>>>
>>>Probability is defined as a mathematical limit, as N
>>>approaches infinity.
>>>
>>>That uses a limit as a finite set increases in size - not
>>>a fraction of an infinite set.
>>>
>>>E.g. see:
>>>
>>>http://www.wordiq.com/definition/Probability
>>
>>This doesn't show that probability is defined as a
>>limit - the nearest you get is in the section
>>"Probability in mathematics", where they use "one
>>approach" to give an interpretation - essentially, the
>>frequentist approach. Note that when they discuss
>>Kolmonogorov's definition of probability as a measure,
>>they make no mention of any limits.
>
>
> You *can* define probabilities in terms of limits -
> without reference to infinite sets.
>
You can, but you don't have to. Kolmonogorov didn't, and
that's the formal definition used nowadays in probability
theory. The wordiq.com website includes an article on
Kolmonogorov's formalisation of probability theory.

> Simply beacuse ratios of the sizes of infinite sets make
> little mathematical sense, that does not render all
> notions of probability useless.
>
You are claiming this, but I have yet to see any proof.
Kolmonogorov's formulation of probability works fine for
(countably) infinite sets of events, and hence there
have to a be (countably) infinite number of probablities
in the measure (i.e. the probability that 1, 2, 3,...
events occur).

>
>>>>I have a colleague who even wrote mathematical papers
>>>>about fractions of uncountable sets.
>>>
>>>If you can show me, I should be able to tell you if they
>>>contain the fallacy under discussion.
>>>
>>>Probably he doesn't do that at all - and instead uses
>>>a limit.
>>
>>This was (I think - my copy is at home) the paper:
>>
>>E. Arjas & E. Nummelin & R.L. Tweedie: Semi-Markov
>> processes on a general state space -theory and quasi-
>> stationarity. J. Aust. Math. Soc. (Series A) 30 (1980):
>> 187 - 200.
>
>
> Apparently too inaccessible for me to examine on the basis
> of a esoteric point in a usenet debate - unless you know
> where it is publicly accessible.
>
Alas not.

>>>Are you suggesting I don't know what I am talking about?
>>>
>>>That is not the case.
>>
>>Your evidence for this is?
>
>
> My mathematical credentials may not be publicly accessible
> for your inspection - but I do have a degree in
> mathematics.
>
> Here is the (basically correct) answer given on mathforum
> regarding ratios of infinite quantities.
>
> http://mathforum.org/library/drmath/view/53337.html
>
And he agrees with me:

"There are sort of some different "sizes" of infinities, so
this means that a quotient that looks like infinity over
infinity can sometimes be a real number, and sometimes it is
just infinity."

He claims that infinity/infinity _can_ be a real number.

Bob

--
Bob O'Hara

Dept. of Mathematics and Statistics
P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of
Helsinki Finland Telephone: +358-9-191 23743 Mobile:
+358 50 599 0540 Fax: +358-9-191 22 779 WWW:
http://www.RNI.Helsinki.FI/~boh/ Journal of Negative
Results - EEB: http://www.jnr-eeb.org
 
John Edser wrote:
>>>Maybe there is some magical method by which the random
>>>part can be extracted after the fact, but that seems
>>>unlikely - only a part of the result that meets tests for
>>>randomness which is likely to be a different matter
>>>altogether.
>>
>
>>Many of the more popular myths of the current paradigm of
>>evolutionary biology pivot off a kind of rhetorical trick.
>>Specifically the trick involves employing a word that has
>>more than one meaning in an argument (or special case) to
>>achieve the illusion of scientific validity. This is *all*
>>that's going on with the Hardy-Weinberg, socalled, Law.
>>And you hit the nail on the head with respect to which
>>word is the "pivot" with respect to how this rhetorical
>>trick is manifested in Hardy-Weinberg: randomness.
>
>
> BOH:- Wierd. The Hardy-Weinberg law is deterimistic: there
> is no randomness in it.
>
> JE:- "Wierd"? Dr O'Hara is not correct. The HW
> distribution is just a binomial distribution derived from
> Pascale's Triangle.

Sorry, John, but there's simply no such thing as the "HW
distribution" in the Hardy-Weinberg law. If I'm wrong, then
please correct me by pointing to the literature which places
the "HW distribution" in the H-W law.

Bob

--
Bob O'Hara

Dept. of Mathematics and Statistics
P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of
Helsinki Finland Telephone: +358-9-191 23743 Mobile:
+358 50 599 0540 Fax: +358-9-191 22 779 WWW:
http://www.RNI.Helsinki.FI/~boh/ Journal of Negative
Results - EEB: http://www.jnr-eeb.org
 
>>JE:- "Wierd"? Dr O'Hara is not correct. The HW
>>distribution is just a binomial distribution derived from
>>Pascale's Triangle.

> BOH:- Sorry, John, but there's simply no such thing as the
> "HW distribution" in the Hardy-Weinberg law. If I'm wrong,
> then please correct me by pointing to the literature which
> places the "HW distribution" in the H-W law.

> JE:- Pedantic nonsense. Do you deny or confirm that HW
> “law” only represents a binomial _distribution_ of the
> alleles concerned so the HW “law” is actually just a
> binomial distribution of alleles, i.e. HW “law” only
> represents a defined _zero_ state re: any _significant_
> bias of a binomial distribution of the alleles concerned?

BOH:- Clearly not: (1) H-W is deterministic, there is no
distribution, (2) the binomial distribution only has two
classes of events: "success" and "failure". H-W has three:
two homozygotes and a heterozygote.

JE:- These "three classes" of events are part and parcel of
the binomial distribution pattern evident within Pascale's
Triangle. If they were not, then they could not be
calculated using a binomial distribution.

A binomial distribution is "deterministic" but the only
thing that matters here is WHAT is it determining? It is
determining the zero state for the distribution of alleles
in homozygous and heterozygous states within a heuristic
infinite population with random mating. This is equivalent
to a random distribution of these alleles within the
supposed population. HW simply predicts the freq of alleles
as ****/hetero zygotes when nothing of significance is
happening to skew the binomial distribution. It just
represents a population geneticists _defined_, blank sheet.

Respectfully,

John Edser Independent Researcher

PO Box 266 Church Pt Australia

[email protected]
 
> >>JE:- "Wierd"? Dr O'Hara is not correct. The HW
> >>distribution is just a binomial distribution derived
> >>from Pascale's Triangle.

> > BOH:- Sorry, John, but there's simply no such thing as
> > the "HW distribution" in the Hardy-Weinberg law. If I'm
> > wrong, then please correct me by pointing to the
> > literature which places the "HW distribution" in the
> > H-W law.

> > JE:- Pedantic nonsense. Do you deny or confirm that HW
> > “law” only represents a binomial _distribution_ of the
> > alleles concerned so the HW “law” is actually just a
> > binomial distribution of alleles, i.e. HW “law” only
> > represents a defined _zero_ state re: any _significant_
> > bias of a binomial distribution of the alleles
> > concerned?

> BOH:- Clearly not: (1) H-W is deterministic, there is no
> distribution, (2) the binomial distribution only has two
> classes of events: "success" and "failure". H-W has three:
> two homozygotes and a heterozygote.

JM:- John, The "binomial distribution" is actually a two
parameter family of distributions. The parameters are "p"
and "n". For n=2, the distribution is, as you claim, very
closely related to Hardy- Weinberg.

JE:- Yes, a binomial distribution is just, Pascale's
Triangle which represents a maximal possible increase. The
only thing than can disturb this prediction of allele
distribution is something that reduces this maximum,
e.g. selection on something.

JM:- I have no idea why BOH claims that the binomial
distribution has only two classes of events. My texts
indicate that it has n+1 classes.

JE:- BOH appears to be attempting to evade the issue which
is the HW "law" only represents a DEFINED zero state for the
distribution of alleles in either the heterozygous or
homozygous state.

JM:- Nor do I understand what he means by the law being
"deterministic".

JE:- A binomial distribution is a determined maximal
increase. However in this case it only determines a zero
state for allele distribution.

JM:- I would recommend that you disengage from discussing
these subjects with BOH. He is apparently peeved with
amateurs who have not yet learned the jargon. Darwin is
quoted as having complained to the effect that when someone
is slain by mathematics, they don't even get the
satisfaction of knowing how they were killed. BOH seems to
be determined not to provide satisfaction. That, apparently,
provides him with satisfaction.

JE:- I am not afraid of the big bad wolf ;-)

Respectfully,

John Edser Independent Researcher

PO Box 266 Church Pt NSW 2105 Australia

[email protected]
 
William L Hunt wrote:
> On Tue, 22 Jun 2004 20:16:27 +0000 (UTC), "Anon."
> <[email protected]> wrote:
>
>
>>William L Hunt wrote:
>>
>>>On Sat, 19 Jun 2004 22:40:58 +0000 (UTC), "Anon."
>>><[email protected]> wrote:
>>
>
>
>>>>>Popularisers should make explicit the behaviour is what
>>>>>happens as the population size tends towards infinity -
>>>>>and not attempt to pass it off as an effect in an
>>>>>infinite population.
>>>>
>
>>>>:-BOH
>>>>But it is - in finite populations, you get an excess of
>>>>homozygotes, as any student of population genetics
>>>>should know.
>>>
>>>:-WLH
>>> This above statement doesn't sound right to me? It is
>>> known that if the matings are other than random there
>>> will be an excess of homozygotes (over a Hardy-Weinberg
>>> equilibrium prediction with random matings) but this is
>>> even usually best to see when the populations are
>>> large. Small populations (with random matings) are
>>> expected to diverge from a precise Hardy-Weinberg
>>> equilibrium simply from the sampling effects of the
>>> small size but I don't recall any bias to this
>>> divergence (more or fewer homozygotes than a Hardy-
>>> Weinberg prediction). If the sampling (mating) is truly
>>> random, I don't see how you could predict a direction
>>> (excess of homozygotes)?
>>
>>:-BOH
>>I don't know which textbooks you have to hand, I have
>>Futuyma's "Evolutionary Biology" (2nd ed. from 1986), and
>>in Chapter 5 ("Population Structure and Genetic Drift") he
>>has a section called "Population Size, Inbreeding, and
>>Genetic Drift" where he shows that any finite population
>>will become inbred, which means a reduction in
>>heterozygosity. I'm sure the same thing is in Hartl &
>>Clarke. Look out for equations like H_t = H_0 (1-1/2N)^t.
>>
>>In essence, any finite population will become inbred over
>>time (at least to some extent), and this increases
>>homozygosity.
>>
>>Bob
>>
>
>
> The discussion was of a Hardy-Weinberg equilibrium and
> your original statement "in finite populations, you get
> an excess of homozygotes", I took to refer to an excess
> over that predicted by HW. Apparently you were referring
> to something else that has nothing to do with Hardy-
> Weinberg equilibrium. This threw me off and I suspect it
> may have for some others also. Guy Hoelzer also responds
> to this in an above thread. Yes, if the population is
> small, many more loci will reach fixation (homozygote) to
> one allele or the other and there is less genetic
> diversity. This has nothing to do with Hardy-Weinberg.
> Hardy-Weinbery only speaks to loci where there are still
> two alleles present in the population at some frequency
> and it predicts what the distribution will be. It
> originally sounded like you were saying was that if there
> is a frequency of p=.5 for allele 'A' and q=.5 for allele
> 'a', that in a large population you would expect the
> distribution to be a Hardy-Weinberg AA=.25, Aa = .50,
> aa=.25 but for some small population size, "you" might
> expect it show an "excess of homozygotes", such as
> AA=.30, Aa=.40, aa=.30. I now am not sure what you meant?

This is what I meant. In a finite population, the expected
number of heterozygotes is less than predicted from HWE.
Gale goes through the calculations in his textbook (my
edition is from the early 80s). Most textbooks use a
deterministic calculation, but get the same result. Either
way, it all goes back to Wright.

Bob

--
Bob O'Hara

Dept. of Mathematics and Statistics
P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of
Helsinki Finland Telephone: +358-9-191 23743 Mobile:
+358 50 599 0540 Fax: +358-9-191 22 779 WWW:
http://www.RNI.Helsinki.FI/~boh/ Journal of Negative
Results - EEB: http://www.jnr-eeb.org
 
On Mon, 28 Jun 2004 16:02:21 +0000 (UTC), "Anon."
<[email protected]> wrote:

>William L Hunt wrote:
>> On Tue, 22 Jun 2004 20:16:27 +0000 (UTC), "Anon."
>> <[email protected]> wrote:
>>
>>
>>>William L Hunt wrote:
>>>
>>>>On Sat, 19 Jun 2004 22:40:58 +0000 (UTC), "Anon."
>>>><[email protected]> wrote:
>>>
>>
>>
>>>>>>Popularisers should make explicit the behaviour is
>>>>>>what happens as the population size tends towards
>>>>>>infinity - and not attempt to pass it off as an effect
>>>>>>in an infinite population.
>>>>>
>>
>>>>>:-BOH
>>>>>But it is - in finite populations, you get an excess of
>>>>>homozygotes, as any student of population genetics
>>>>>should know.
>>>>
>>>>:-WLH
>>>> This above statement doesn't sound right to me? It is
>>>> known that if the matings are other than random there
>>>> will be an excess of homozygotes (over a Hardy-
>>>> Weinberg equilibrium prediction with random matings)
>>>> but this is even usually best to see when the
>>>> populations are large. Small populations (with random
>>>> matings) are expected to diverge from a precise Hardy-
>>>> Weinberg equilibrium simply from the sampling effects
>>>> of the small size but I don't recall any bias to this
>>>> divergence (more or fewer homozygotes than a Hardy-
>>>> Weinberg prediction). If the sampling (mating) is
>>>> truly random, I don't see how you could predict a
>>>> direction (excess of homozygotes)?
>>>
>>>:-BOH
>>>I don't know which textbooks you have to hand, I have
>>>Futuyma's "Evolutionary Biology" (2nd ed. from 1986), and
>>>in Chapter 5 ("Population Structure and Genetic Drift")
>>>he has a section called "Population Size, Inbreeding, and
>>>Genetic Drift" where he shows that any finite population
>>>will become inbred, which means a reduction in
>>>heterozygosity. I'm sure the same thing is in Hartl &
>>>Clarke. Look out for equations like H_t = H_0 (1-1/2N)^t.
>>>
>>>In essence, any finite population will become inbred over
>>>time (at least to some extent), and this increases
>>>homozygosity.
>>>
>>>Bob
>>>
>>
>> :-WLH
>> The discussion was of a Hardy-Weinberg equilibrium and
>> your original statement "in finite populations, you get
>> an excess of homozygotes", I took to refer to an excess
>> over that predicted by HW. Apparently you were referring
>> to something else that has nothing to do with Hardy-
>> Weinberg equilibrium. This threw me off and I suspect it
>> may have for some others also. Guy Hoelzer also responds
>> to this in an above thread. Yes, if the population is
>> small, many more loci will reach fixation (homozygote)
>> to one allele or the other and there is less genetic
>> diversity. This has nothing to do with Hardy-Weinberg.
>> Hardy-Weinbery only speaks to loci where there are still
>> two alleles present in the population at some frequency
>> and it predicts what the distribution will be. It
>> originally sounded like you were saying was that if
>> there is a frequency of p=.5 for allele 'A' and q=.5 for
>> allele 'a', that in a large population you would expect
>> the distribution to be a Hardy-Weinberg AA=.25, Aa =
>> .50, aa=.25 but for some small population size, "you"
>> might expect it show an "excess of homozygotes", such as
>> AA=.30, Aa=.40, aa=.30. I now am not sure what you
>> meant?
>
>:-BOH
>This is what I meant. In a finite population, the expected
>number of heterozygotes is less than predicted from HWE.
>Gale goes through the calculations in his textbook (my
>edition is from the early 80s). Most textbooks use a
>deterministic calculation, but get the same result. Either
>way, it all goes back to Wright.
>
>Bob
>
Can you give me a formula that for a particular locus,
given N q and p for a randomly mating population, gives the
expected frequency of AA,Aa, and aa similiar to HWE above
but allows one to see what distribution is expected for a
particular finite population size(N)? I don't recall seeing
such a formula? William L Hunt

... [snip] ...
 
Guy Hoelzer <[email protected]> wrote or quoted:
> Tim Tyler at [email protected] wrote on 6/23/04 9:06 AM:
> > Perplexed in Peoria <[email protected]> wrote or
> > quoted:
> >> In reply to a post from Bob containing:

> >>> I don't know which textbooks you have to hand, I have
> >>> Futuyma's "Evolutionary Biology" (2nd ed. from 1986),
> >>> and in Chapter 5 ("Population Structure and Genetic
> >>> Drift") he has a section called "Population Size,
> >>> Inbreeding, and Genetic Drift" where he shows that any
> >>> finite population will become inbred, which means a
> >>> reduction in heterozygosity. [...]
> >>
> >>> In essence, any finite population will become inbred
> >>> over time (at least to some extent), and this
> >>> increases homozygosity.
> >>
> >> Boy, are you going to feel foolish after you get a good
> >> night's sleep and review what you have written :)
> >
> > I believe it is conventional enough.
> >
> > A finite population will be subject to drift - which
> > will remove alleles from the population - and increase
> > homozygosity as a result.
> >
> > Ridley also refers to this effect as "inbreeding" -
> > writing in his textbook:
> >
> > "The increase in homozygosity under drift is due to
> > inbreeding".
>
> Yeah. This is the kind of statement that led me to stop
> using this textbook in my evolution course. Inbreeding
> does not cause drift, or vice versa. The strength of drift
> changes inversely with effective population size (by
> definition) and inbreeding tends to vary the same way.

A "sufficiently-inbred" section of a population could well
be indistinguishable from a population consisting of a small
number of individuals.

In that sense it would be reasonable to say that a
particular kind of inbreeding could be regarded as being
solely responsible for all the effects of small
population sizes.

Incidentally - I *like* the textbook - along with most of
the other stuff Matt Ridley writes - and I expect I'll
continue to refer to it in glowing terms - at least until
I'm exposed to something similar - but significantly better.
--
__________
|im |yler http://timtyler.org/ [email protected] Remove
lock to reply.
 
William L Hunt wrote:
> On Mon, 28 Jun 2004 16:02:21 +0000 (UTC), "Anon."
> <[email protected]> wrote:
>
>
>>William L Hunt wrote:
>>
>>>On Tue, 22 Jun 2004 20:16:27 +0000 (UTC), "Anon."
>>><[email protected]> wrote:
>>>
>>>
>>>
>>>>William L Hunt wrote:
>>>>
>>>>
>>>>>On Sat, 19 Jun 2004 22:40:58 +0000 (UTC), "Anon."
>>>>><[email protected]> wrote:
>>>>
>>>
>>>>>>>Popularisers should make explicit the behaviour is
>>>>>>>what happens as the population size tends towards
>>>>>>>infinity - and not attempt to pass it off as an
>>>>>>>effect in an infinite population.
>>>>>>
>>>>>>:-BOH
>>>>>>But it is - in finite populations, you get an excess
>>>>>>of homozygotes, as any student of population genetics
>>>>>>should know.
>>>>>
>>>>>:-WLH
>>>>> This above statement doesn't sound right to me? It is
>>>>> known that if the matings are other than random there
>>>>> will be an excess of homozygotes (over a Hardy-
>>>>> Weinberg equilibrium prediction with random matings)
>>>>> but this is even usually best to see when the
>>>>> populations are large. Small populations (with random
>>>>> matings) are expected to diverge from a precise Hardy-
>>>>> Weinberg equilibrium simply
>>>>
>>>>>from the sampling effects of the small size but I don't
>>>>>recall any
>>>>
>>>>>bias to this divergence (more or fewer homozygotes than
>>>>>a Hardy-Weinberg prediction). If the sampling (mating)
>>>>>is truly random, I don't see how you could predict a
>>>>>direction (excess of homozygotes)?
>>>>
>>>>:-BOH
>>>>I don't know which textbooks you have to hand, I have
>>>>Futuyma's "Evolutionary Biology" (2nd ed. from 1986),
>>>>and in Chapter 5 ("Population Structure and Genetic
>>>>Drift") he has a section called "Population Size,
>>>>Inbreeding, and Genetic Drift" where he shows that any
>>>>finite population will become inbred, which means a
>>>>reduction in heterozygosity. I'm sure the same thing is
>>>>in Hartl & Clarke. Look out for equations like H_t = H_0
>>>>(1-1/2N)^t.
>>>>
>>>>In essence, any finite population will become inbred
>>>>over time (at least to some extent), and this increases
>>>>homozygosity.
>>>>
>>>>Bob
>>>>
>>>
>>>:-WLH
>>> The discussion was of a Hardy-Weinberg equilibrium and
>>> your original statement "in finite populations, you get
>>> an excess of homozygotes", I took to refer to an excess
>>> over that predicted by HW. Apparently you were referring
>>> to something else that has nothing to do with Hardy-
>>> Weinberg equilibrium. This threw me off and I suspect it
>>> may have for some others also. Guy Hoelzer also responds
>>> to this in an above thread. Yes, if the population is
>>> small, many more loci will reach fixation (homozygote)
>>> to one allele or the other and there is less genetic
>>> diversity. This has nothing to do with Hardy-Weinberg.
>>> Hardy-Weinbery only speaks to loci where there are still
>>> two alleles present in the population at some frequency
>>> and it predicts what the distribution will be. It
>>> originally sounded like you were saying was that if
>>> there is a frequency of p=.5 for allele 'A' and q=.5 for
>>> allele 'a', that in a large population you would expect
>>> the distribution to be a Hardy-Weinberg AA=.25, Aa =
>>> .50, aa=.25 but for some small population size, "you"
>>> might expect it show an "excess of homozygotes", such as
>>> AA=.30, Aa=.40, aa=.30. I now am not sure what you
>>> meant?
>>
>>:-BOH
>>This is what I meant. In a finite population, the expected
>>number of heterozygotes is less than predicted from HWE.
>>Gale goes through the calculations in his textbook (my
>>edition is from the early 80s). Most textbooks use a
>>deterministic calculation, but get the same result. Either
>>way, it all goes back to Wright.
>>
>>Bob
>>
>
> Can you give me a formula that for a particular locus,
> given N q and p for a randomly mating population, gives
> the expected frequency of AA,Aa, and aa similiar to HWE
> above but allows one to see what distribution is expected
> for a particular finite population size(N)? I don't
> recall seeing such a formula?

I suspect you have seen it, but not expressed in this way!

The expected frequency of heterozygotes is 2pq(1-1/2N), the
expected frequency of the AA homozygote is p^2 + p(1-p)/2N.
It turns out that
1/2N is just the inbreeding coefficient, of course.

My reference is J. S. Gale (1980) Population Genetics
(Tertiary Level Biology), and I forgot to bring it in to
work today. But the proof of the heterozygote deficit
follwos from calculating E[2p(1-p)] = 2E[p] - 2E[p^2].

E[p]=p and E[p^2] is calculated from the definition of
a variance:

Var[p] = E[p^2] - E^2[p] and Var[p] = p(1-p)/2N, so E[p^2] =
p(1-p)/2N - p^2.

Plugging this into E[2p(1-p)] we get

2E[p] - 2E[p^2] = 2p - 2( p(1-p)/2N - p^2 )
= 2(p-p^2) - 2p(1-p)/2N 2p(1-p)(1-1/2N)

which is the result we want.

Bob

--
Bob O'Hara

Dept. of Mathematics and Statistics
E.q. Box 4 (Yliopistonkatu 5) FIN-00014 University of
Helsinki Finland Telephone: +358-9-191 23743 Mobile:
+358 50 599 0540 Fax: +358-9-191 22 779 WWW:
http://www.RNI.Helsinki.FI/~boh/ Journal of Negative
Results - EEB: http://www.jnr-eeb.org
 
"Tim Tyler" <[email protected]> wrote in message
news:[email protected]...

> You cannot simply talk about ratios between two infinite
> quantities.

Agreed.

> so - unqualified mention of allele frequency in an
> infinite population makes no sense.

Your inference is valid only if you make the auxiliary
assumption that "frequency" is defined using a ratio. It
need not be. It can be defined directly as a probability.
Then ratios come in (via the law of large numbers) only
indirectly and only for finite samples or populations.

Returning to Hardy-Weinberg, you will notice that the
derivation of the law uses frequencies as probabilities
(i.e. random mating) - ratios never enter into it, since
large quantities N*p never come into it.

However, in your current discussion with BOH regarding
whether half of the natural numbers are even, you should be
able to prevail easily enough. He doesn't seem to have
grasped the subtleties. He did have a great pun about
"Apology" though.
 
Anon. <[email protected]> wrote or quoted:
> Tim Tyler wrote:
> > Anon. <[email protected]> wrote or
> > quoted:

> >>>>>It's like claiming that half the integers are even.
> >>>>
> >>>>Err, they are. There are just rather a lot of them.
> >>>
> >>>No, there aren't.
> >>>
> >>>There are an infinite number of even numbers.
> >>>
> >>>There are an infinite number of odd numbers.
> >>>
> >>>Divide infinity by infinity and the result is
> >>>indeterminate.
> >>
> >>If there are an equal number of even and odd numbers,
> >>then half of the numbers must be even.
> >
> > This is not true when the sizes of the sets involved are
> > infinite.
>
> But they're the same size! We can count them!

That doesn't help - since the sizes are infinite - and one
infinite number divided by another one does not necessarily
equal 0.5.

> >>This must be true because for every even number, I can
> >>add 1 and get an odd number. Conversely for every odd
> >>number I can add 1 and get an even number. Hence, by the
> >>operation of adding 1, I can produce an even number for
> >>every odd number and vice versa. Ergo, half of all
> >>numbers are even, and half are odd.
> >
> > I can easily create a map between every even number an 5
> > unique odd numbers - i.e I can map from 2x to 5x+1,
> > 5x+3, 5x+5, 5x*7 and 5x+9.
>
> Yes, but that's not the operation of adding 1 is it?

I never said it was.

What it proves that - as well as there being one odd number
for every even number there are also five odd numbers for
every even number.

That's not good news for the assertion that the ratio of the
number of even numbers to the number of odd numbers is one.
Much the same argument will "prove" the ratio is anything
you care to mention.

> What I missed out was a uniqueness statement (i.e. that I
> can create all odd numbers uniquely from the even
> numbers).

My map is the same - all the odd numbers are generated
uniquely from the set of even numbers.

> > That is exactly the same sort of argument as the one you
> > gave - yet it indicates that there are *five* odd
> > numbers for every even number.
> >
> > What conclusion should one draw from this?
> >
> > The correct conclusion is that the argument you gave is
> > useless.
>
> Not quite, because I can resurrect it. [...]

IMO, the argument on this point is *totally* dead to the
point where it is not worth discussing any more -
especially not here.

> >>>>>No serious mathematician can talk about fractions of
> >>>>>infinite sets and expect to be taken seriously.
> >>>>
> >>>>But they do.
> >>>
> >>>No - not unless the fractions are "zero" or "one".
> >>
> >>Rubbish, unless you're denying the existence of
> >>fractions. Fractions are fractions of an infinite set,
> >>because there is an infinite number of numbers between 0
> >>and 1 (proof: take the reciprocal of every positive
> >>integer).
> >
> > 1/3 is *not* the ratio of the size of the set of numbers
> > smaller than 1/3 and the size of the set of numbers
> > greater than 1/3. That ratio is the ratio of two
> > infinite numbers - and thus is not well defined.
>
> So you claim, but have yet to provide a proof. This is
> mathematics, so you should be able to provide a proof of
> your assertion that one can never determine the value of
> the ratio infinity/infinity.

You *could* define it to mean something - the same way that
you define anything else in mathematics - with axioms.

However there is no generally-accepted way of defining it.

You *could* define it using a limit (which I've already said
I'm perfectly happy with) - but then you would *have* to
state which limit you are using - since different limits
typically produce different answers.

> >>>>It's how probability is defined as a concept.
> >>>
> >>>Probability is defined as a mathematical limit, as N
> >>>approaches infinity.
> >>>
> >>>That uses a limit as a finite set increases in size -
> >>>not a fraction of an infinite set.
> >>>
> >>>E.g. see:
> >>>
> >>>http://www.wordiq.com/definition/Probability
> >>
> >>This doesn't show that probability is defined as a limit
> >>- the nearest you get is in the section "Probability in
> >>mathematics", where they use "one approach" to give an
> >>interpretation - essentially, the frequentist approach.
> >>Note that when they discuss Kolmonogorov's definition of
> >>probability as a measure, they make no mention of any
> >>limits.
> >
> > You *can* define probabilities in terms of limits -
> > without reference to infinite sets.
>
> You can, but you don't have to. [...]

So much for your assertion that fractions and probability
depended on ratios between infinite sets:

``by that argument, you can't even define a fraction or a
probability.''

> > Simply beacuse ratios of the sizes of infinite sets make
> > little mathematical sense, that does not render all
> > notions of probability useless.
>
> You are claiming this, but I have yet to see any
> proof. [...]

Probably because this is sci.bio.evolution :-(

If you are *still* in doubt, look up:

"Classical definition of probability"

...and...

"Frequency definition of probability"

...or more simply, just take my word for it that
probability can be quite constently defined as a limit as
the number of samples or trials tends to infinity - and let
the matter drop.

[...]

> > Here is the (basically correct) answer given on
> > mathforum regarding ratios of infinite quantities.
> >
> > http://mathforum.org/library/drmath/view/53337.html
>
> And he agrees with me:
>
> "There are sort of some different "sizes" of infinities,
> so this means that a quotient that looks like infinity
> over infinity can sometimes be a real number, and
> sometimes it is just infinity."
>
> He claims that infinity/infinity _can_ be a real number.

Notice that is next to where he mentions "L'Hopital's Rule"
and "the limit as x approaches infinity".

I said from the start that using limits was permissable -
provided you said what they were.

The problems arise when you talk about ratios of infinite
quantities *without* specifying that what you are /really/
talking about is the result of using a limit or a particular
sampling strategy.

In particular, you can't talk about allele frequencies in an
infinite population *without* mentioning limits, sampling
strategies - or something equivalent - since you could be
talking about anything.

I have to say that "there are sort of some different 'sizes'
of infinities" is misleading nonsense - when it is not
referring to infinities of different cardinalities (as it is
not here).

Alas, this sort of material rather discredits the source
I cited :-|
--
__________
|im |yler http://timtyler.org/ [email protected] Remove
lock to reply.
 
"Anon." <[email protected]> wrote in message news:<[email protected]>...
> Tim Tyler wrote:
> > The correct conclusion is that the argument you gave is
> > useless.
> >
> Not quite, because I can resurrect it. For each integer
> (for which there are a countably finite number), I can
> create a single unique even number (i.e. I multiply the
> integer by 2). I can also create a single unique odd
> number: multiply by 2 and add 1. There are therefore as
> many even numbers as integers, and as many odd numbers as
> integers, so half of the integers must be even.

Tim was probably wise to bow out at this point. I am less
wise. Therefore, I can't resist pointing out that one can
construct a bijection between the natural numbers and the
prime numbers. One can also construct a bijection between
the natural numbers and the composites. One fairly orthodox
conclusion that can be drawn from this is that the number of
primes and the number of composites is the same ("aleph
nought"). Your conclusion that half of the natural numbers
are prime and half composite would be unorthodox and
potentially paradoxical.

Warning! I may also learn wisdom if this subject shows
tendencies to persist.
 
Bob (and Bill),

in article [email protected], Anon. at
[email protected] wrote on
6/30/04 8:35 AM:

>>>> :-WLH
>>>> The discussion was of a Hardy-Weinberg equilibrium and
>>>> your original statement "in finite populations, you get
>>>> an excess of homozygotes", I took to refer to an excess
>>>> over that predicted by HW. Apparently you were
>>>> referring to something else that has nothing to do with
>>>> Hardy-Weinberg equilibrium. This threw me off and I
>>>> suspect it may have for some others also. Guy Hoelzer
>>>> also responds to this in an above thread. Yes, if the
>>>> population is small, many more loci will reach fixation
>>>> (homozygote) to one allele or the other and there is
>>>> less genetic diversity. This has nothing to do with Hardy-
>>>> Weinberg. Hardy-Weinbery only speaks to loci where
>>>> there are still two alleles present in the population
>>>> at some frequency and it predicts what the distribution
>>>> will be. It originally sounded like you were saying was
>>>> that if there is a frequency of p=.5 for allele 'A' and
>>>> q=.5 for allele 'a', that in a large population you
>>>> would expect the distribution to be a Hardy-Weinberg
>>>> AA=.25, Aa = .50, aa=.25 but for some small population
>>>> size, "you" might expect it show an "excess of
>>>> homozygotes", such as AA=.30, Aa=.40, aa=.30. I now am
>>>> not sure what you meant?
>>>
>>> :-BOH
>>> This is what I meant. In a finite population, the
>>> expected number of heterozygotes is less than predicted
>>> from HWE. Gale goes through the calculations in his
>>> textbook (my edition is from the early 80s). Most
>>> textbooks use a deterministic calculation, but get the
>>> same result. Either way, it all goes back to Wright.
>>>
>>> Bob
>>>
>>
>> Can you give me a formula that for a particular locus,
>> given N q and p for a randomly mating population, gives
>> the expected frequency of AA,Aa, and aa similiar to HWE
>> above but allows one to see what distribution is
>> expected for a particular finite population size(N)? I
>> don't recall seeing such a formula?

I was looking forward to Bob's answer, because his claim
seemed so clearly false and that would be out of character.

> I suspect you have seen it, but not expressed in this way!
>
> The expected frequency of heterozygotes is 2pq(1-1/2N),
> the expected frequency of the AA homozygote is p^2 + p(1-
> p)/2N. It turns out that
> 1/2N is just the inbreeding coefficient, of course.

Indeed it is; and now I see the confusion. This set of
equations is used to predict genotype frequencies in the
next generation after a round of random mating. Writing the
equations this way is a little misleading, because the role
of the factors including the "N" term is to account for
changes in "p" and "q" resulting from drift in a finite
population. So, in effect, "p" and "q" drift, eventually
leading to fixation of one or the other allele. Under the
neutral model, we expect a particular time to fixation
(contingent upon p, q, and N; and with a huge variance), and
the equations above predict the proportional change in
genotype frequencies as if inbreeding in the randomly mating
finite population pushed the population deterministically
and without variance to fixation in exactly the expected
number of generations. Of course, drift does not work that
way. Instead, it causes the gene pool to take a random walk
to fixation, and the pure combinatorial predictions of the
infinite population size version of the HW model will be
unbiased at every step along the way.

> My reference is J. S. Gale (1980) Population Genetics
> (Tertiary Level Biology), and I forgot to bring it in to
> work today. But the proof of the heterozygote deficit
> follwos from calculating E[2p(1-p)] = 2E[p] - 2E[p^2].
>
> E[p]=p and E[p^2] is calculated from the definition of a
> variance:
>
> Var[p] = E[p^2] - E^2[p] and Var[p] = p(1-p)/2N, so E[p^2]
> = p(1-p)/2N - p^2.
>
> Plugging this into E[2p(1-p)] we get
>
> 2E[p] - 2E[p^2] = 2p - 2( p(1-p)/2N - p^2 )
> = 2(p-p^2) - 2p(1-p)/2N 2p(1-p)(1-1/2N)
>
> which is the result we want.

It is indeed a valid equation, though prone to a great deal
of stochastic error, for the mean field expectation of
genotype frequency changes across generations in a finite
population.

Cheers,

Guy
 
"Anon." <[email protected]> wrote in message
news:[email protected]...
> William L Hunt wrote:
> > On Mon, 28 Jun 2004 16:02:21 +0000 (UTC), "Anon."
> > <[email protected]> wrote:
> >
> >
> >>William L Hunt wrote:
> >>
> >>>On Tue, 22 Jun 2004 20:16:27 +0000 (UTC), "Anon."
> >>><[email protected]> wrote:
> >>>
> >>>
> >>>
> >>>>William L Hunt wrote:
> >>>>
> >>>>
> >>>>>On Sat, 19 Jun 2004 22:40:58 +0000 (UTC), "Anon."
> >>>>><[email protected]> wrote:
> >>>>
> >>>
> >>>>>>>Popularisers should make explicit the behaviour is
> >>>>>>>what happens as the population size tends towards
> >>>>>>>infinity - and not attempt to
pass
> >>>>>>>it off as an effect in an infinite population.
> >>>>>>
> >>>>>>:-BOH
> >>>>>>But it is - in finite populations, you get an excess
> >>>>>>of homozygotes,
as
> >>>>>>any student of population genetics should know.
> >>>>>
> >>>>>:-WLH
> >>>>> This above statement doesn't sound right to me? It
> >>>>> is known that if the matings are other than random
> >>>>> there will be an excess of homozygotes (over a Hardy-
> >>>>> Weinberg equilibrium
prediction
> >>>>>with random matings) but this is even usually best to
> >>>>>see when the populations are large. Small populations
> >>>>>(with random matings) are expected to diverge from a
> >>>>>precise Hardy-Weinberg equilibrium simply
> >>>>
> >>>>>from the sampling effects of the small size but I
> >>>>>don't recall any
> >>>>
> >>>>>bias to this divergence (more or fewer homozygotes
> >>>>>than a Hardy-Weinberg prediction). If the sampling
> >>>>>(mating) is truly random, I don't see how you could
> >>>>>predict a direction (excess of
homozygotes)?
> >>>>
> >>>>:-BOH
> >>>>I don't know which textbooks you have to hand, I have
> >>>>Futuyma's "Evolutionary Biology" (2nd ed. from 1986),
> >>>>and in Chapter 5 ("Population Structure and Genetic
> >>>>Drift") he has a section called "Population Size,
> >>>>Inbreeding, and Genetic Drift" where he shows that
any
> >>>>finite population will become inbred, which means a
> >>>>reduction in heterozygosity. I'm sure the same thing
> >>>>is in Hartl & Clarke. Look
out
> >>>>for equations like H_t = H_0 (1-1/2N)^t.
> >>>>
> >>>>In essence, any finite population will become inbred
> >>>>over time (at
least
> >>>>to some extent), and this increases homozygosity.
> >>>>
> >>>>Bob
> >>>>
> >>>
> >>>:-WLH
> >>> The discussion was of a Hardy-Weinberg equilibrium and
> >>> your original statement "in finite populations, you
> >>> get an excess of homozygotes", I took to refer to an
> >>> excess over that predicted by HW. Apparently you were
> >>> referring to something else that has nothing to do
> >>> with Hardy-Weinberg equilibrium. This threw me off and
> >>> I suspect it may have for some others also. Guy
> >>> Hoelzer also responds to this in an above thread. Yes,
> >>> if the population is small, many more loci will reach
> >>> fixation (homozygote) to one allele or the other and
> >>> there is less genetic diversity. This has nothing to
> >>> do with Hardy-Weinberg. Hardy-Weinbery only speaks to
> >>> loci where there are still two alleles present in the
> >>> population at some frequency and it predicts what the
> >>> distribution will be. It originally sounded like you
> >>> were saying was that if there is a frequency of p=.5
> >>> for allele 'A' and q=.5 for allele 'a', that in a
> >>> large population you would expect the distribution to
> >>> be a Hardy-Weinberg AA=.25, Aa = .50, aa=.25 but for
> >>> some small population size, "you" might expect it show
> >>> an "excess of homozygotes", such as AA=.30, Aa=.40,
> >>> aa=.30. I now am not sure what you meant?
> >>
> >>:-BOH
> >>This is what I meant. In a finite population, the
> >>expected number of heterozygotes is less than predicted
> >>from HWE. Gale goes through the calculations in his
> >>textbook (my edition is from the early 80s). Most
> >>textbooks use a deterministic calculation, but get the
> >>same result. Either way, it all goes back to Wright.
> >>
> >>Bob
> >>
> >
> > Can you give me a formula that for a particular locus,
> > given N q and p for a randomly mating population, gives
> > the expected frequency of AA,Aa, and aa similiar to HWE
> > above but allows one to see what distribution is
> > expected for a particular finite population size(N)? I
> > don't recall seeing such a formula?
>
> I suspect you have seen it, but not expressed in this way!
>
> The expected frequency of heterozygotes is 2pq(1-1/2N),
> the expected frequency of the AA homozygote is p^2 + p(1-
> p)/2N. It turns out that
> 1/2N is just the inbreeding coefficient, of course.
>
> My reference is J. S. Gale (1980) Population Genetics
> (Tertiary Level Biology), and I forgot to bring it in to
> work today. But the proof of the heterozygote deficit
> follwos from calculating E[2p(1-p)] = 2E[p] - 2E[p^2].
>
> E[p]=p and E[p^2] is calculated from the definition of a
> variance:
>
> Var[p] = E[p^2] - E^2[p] and Var[p] = p(1-p)/2N, so E[p^2]
> = p(1-p)/2N - p^2.
>
> Plugging this into E[2p(1-p)] we get
>
> 2E[p] - 2E[p^2] = 2p - 2( p(1-p)/2N - p^2 )
> = 2(p-p^2) - 2p(1-p)/2N 2p(1-p)(1-1/2N)
>
> which is the result we want.

Ah! I think I get it now. To put your argument into my
(probably imprecise) words:

HW predicts expected heterozygote frequency of 2p(1-p)
on the assumption of one generation of random mating
with no drift.

BOH/Gale predicts expected heterozygote frequency of 2p(1-p)(1-
1/2N) on the assumption of one generation of random mating
with one generation of drift.

Since there is really no way to have random mating without
drift, the BOH/Gale assumptions are the correct assumptions
to make in a finite population. And that, rather than the
issue of selfing as I had assumed, is the reason why careful
geneticists endorse the HW predictions only in the limit of
an infinite population.

Now I see why you don't feel foolish. I feel a little
foolish because I had interpreted what you were saying as
being an effect of past drift, rather than a fresh effect of
one generation's worth of drift. ;-( My apologies.

However, I still think you are wrong on
"binomial/trinomial". ;-)
 
Perplexed in Peoria <[email protected]> wrote or quoted:
>
> "Tim Tyler" <[email protected]> wrote in message
> news:[email protected]...
>
> > You cannot simply talk about ratios between two infinite
> > quantities.
>
> Agreed.
>
> > so - unqualified mention of allele frequency in an
> > infinite population makes no sense.
>
> Your inference is valid only if you make the auxiliary
> assumption that "frequency" is defined using a ratio. It
> need not be. It can be defined directly as a probability.
> Then ratios come in (via the law of large numbers) only
> indirectly and only for finite samples or populations.

If you are picking samples from a population and looking at
the limit as the sample size tends to infinity, you *still*
need to specify how you are choosing your sample.

Different approaches to taking samples are likely to lead to
different ratios.

You can't just say something like "sample at random" - since
sampling at random from an infinite population is not
normally a well-defined operation either.

> Returning to Hardy-Weinberg, you will notice that the
> derivation of the law uses frequencies as probabilities
> (i.e. random mating) - ratios never enter into it, since
> large quantities N*p never come into it.

While it makes sense in finite populations, "random mating"
is not a coherent concept in an infinite population. Nor
can you sensibly discuss the probability of organisms
having a particular genome - *unless* you specify a
sampling strategy.

I have no problem with Hardy-Weinberg - provided it is
expressed as a limit as N -> oo.
--
__________
|im |yler http://timtyler.org/ [email protected] Remove
lock to reply.
 
Tim Tyler wrote:
> Anon. <[email protected]> wrote
> or quoted:
>
>>Tim Tyler wrote:
>>
>>>Anon. <[email protected]> wrote or
>>>quoted:
>>
>
>>>>>>>It's like claiming that half the integers are even.
>>>>>>
>>>>>>Err, they are. There are just rather a lot of them.
>>>>>
>>>>>No, there aren't.
>>>>>
>>>>>There are an infinite number of even numbers.
>>>>>
>>>>>There are an infinite number of odd numbers.
>>>>>
>>>>>Divide infinity by infinity and the result is
>>>>>indeterminate.
>>>>
>>>>If there are an equal number of even and odd numbers,
>>>>then half of the numbers must be even.
>>>
>>>This is not true when the sizes of the sets involved are
>>>infinite.
>>
>>But they're the same size! We can count them!
>
>
> That doesn't help - since the sizes are infinite - and one
> infinite number divided by another one does not
> necessarily equal 0.5.
>
>
>>>>This must be true because for every even number, I can
>>>>add 1 and get an odd number. Conversely for every odd
>>>>number I can add 1 and get an even number. Hence, by the
>>>>operation of adding 1, I can produce an even number for
>>>>every odd number and vice versa. Ergo, half of all
>>>>numbers are even, and half are odd.
>>>
>>>I can easily create a map between every even number an 5
>>>unique odd numbers - i.e I can map from 2x to 5x+1, 5x+3,
>>>5x+5, 5x*7 and 5x+9.
>>
>>Yes, but that's not the operation of adding 1 is it?
>
>
> I never said it was.
>
> What it proves that - as well as there being one odd
> number for every even number there are also five odd
> numbers for every even number.
>
> That's not good news for the assertion that the ratio of
> the number of even numbers to the number of odd numbers is
> one. Much the same argument will "prove" the ratio is
> anything you care to mention.
>
I think you're missing my point - that one can create a one
to one correspondence between each even and odd number by
the process of adding 1.

Others seem to think that I'm doing something wrong, and I
may well be, in which case I'd like to know what the problem
is - email to me (not the list)!

<snip>

>>>Simply beacuse ratios of the sizes of infinite sets make
>>>little mathematical sense, that does not render all
>>>notions of probability useless.
>>
>>You are claiming this, but I have yet to see any
>>proof. [...]
>
>
> Probably because this is sci.bio.evolution :-(
>
> If you are *still* in doubt, look up:
>
> "Classical definition of probability"
>
> ...and...
>
> "Frequency definition of probability"
>
> ...or more simply, just take my word for it that
> probability can be quite constently defined as a limit as
> the number of samples or trials tends to infinity - and
> let the matter drop.
>
As long as you admit that you have heard of Kolmonogorov's
definition of probability, which does not depend on a limit
argument. To get back to my original point, this means that
one can define proportions from infinite populations -
Kolmonogorov defines the set that he constructs his measure
so that it can be countably infinite.

Yes, probability (and proportion) _can_ be defined as a
limit, but there are other ways of doing this, and so an
insistence on the necessity of a limit argument is incorrect
- that's the only point I was trying to make.

Bob

--
Bob O'Hara

Dept. of Mathematics and Statistics
P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of
Helsinki Finland Telephone: +358-9-191 23743 Mobile:
+358 50 599 0540 Fax: +358-9-191 22 779 WWW:
http://www.RNI.Helsinki.FI/~boh/ Journal of Negative
Results - EEB: http://www.jnr-eeb.org
 
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