[SOLVED] probability (was Re: EEQT)

Aaron Denney

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\nOn 2004-08-12, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n> Probabilities of single events are meaningless.\n\nOh boy. This is a really blatant misconception of probability. I can\'t\ntell whether the context rescues this (it seems not, though you are\narguing against another \'interpretation\' that I also find odious), but\nthe words themselves make probability a useless concept.\n\nOne can try to rescue it, e.g. by looking at ensembles, assuming\nexchangeability & independence, and trying to identify frequencies with\nprobabilities, but this gets incoherent, circular, or irrelevant to\nexperiment very quickly.\n\nIndependence is a statement about non-correlation between probabilities\nof single events, which you claim are meaningless.\n\nExchangeability is a statement that certain classes of possible\nrealizations of ensembles are equally likely -- in other words, it\ndepends on having a concept of probability for single events, where\nthese single events are realizations of ensemble measurements.\n\nIf you want to use finite ensembles, all probabilities must now be\nrationals, which seems a big limitation. We\'re also not guaranteed that\nthe final frequency really has any connection to the probability as we\nwant it -- instead we have to bring up "for all practical purposes"\narguments.\n\nIf you want to use infinite ensembles (the only case where we can\n_really_ say that the limiting frequencies approach the probability),\nwe have another problem: infinite ensembles\' limiting frequencies are a\ntail property. We can only measure the head, which can be arbitrarily\ndifferent, no matter how far out you measure it. There\'s no grounding\nthat lets us connect the rest of the ensemble to what we actually\nmeasure.\n\nBoth types of ensembles are imaginary and have nothing to do with\nany data we\'ve actually measured.\n\n--\nAaron Denney\n-><-\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On [itex]2004-08-12,[/itex] Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
> Probabilities of single events are meaningless.

Oh boy. This is a really blatant misconception of probability. I can't
tell whether the context rescues this (it seems not, though you are
arguing against another 'interpretation' that I also find odious), but
the words themselves make probability a useless concept.

One can try to rescue it, e.g. by looking at ensembles, assuming
exchangeability & independence, and trying to identify frequencies with
probabilities, but this gets incoherent, circular, or irrelevant to
experiment very quickly.

Independence is a statement about non-correlation between probabilities
of single events, which you claim are meaningless.

Exchangeability is a statement that certain classes of possible
realizations of ensembles are equally likely -- in other words, it
depends on having a concept of probability for single events, where
these single events are realizations of ensemble measurements.

If you want to use finite ensembles, all probabilities must now be
rationals, which seems a big limitation. We're also not guaranteed that
the final frequency really has any connection to the probability as we
want it -- instead we have to bring up "for all practical purposes"
arguments.

If you want to use infinite ensembles (the only case where we can
_really_ say that the limiting frequencies approach the probability),
we have another problem: infinite ensembles' limiting frequencies are a
tail property. We can only measure the head, which can be arbitrarily
different, no matter how far out you measure it. There's no grounding
that lets us connect the rest of the ensemble to what we actually
measure.

Both types of ensembles are imaginary and have nothing to do with
any data we've actually measured.

--
Aaron Denney
[itex]-><-[/itex]

Arnold Neumaier

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nAaron Denney wrote:\n> On 2004-08-12, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n>\n>>Probabilities of single events are meaningless.\n>\n> Oh boy. This is a really blatant misconception of probability.\n\nThe misconception seems to be completely on your side.\n\nWhat is the probability that \'I will die of cancer\'?\nThis is a single event that either will happen, or will not happen.\nIf you consider this single event only, the probability is 1 or 0\ndepending on what will actually happen. (But this sort of probability\nis not what we talk about in physics.)\n\nOn the other hand one may assign a probability based on some facts\nabout me. These facts determine an ensemble of people, from which\none can form a statistical estimate of the probability.\n\nIt clearly depends on which sort of ensemble one regarde me\nto belong to, what probability you will assign.\nI belong to many ensembles, and the answer is different for each\nof these.\n\nThus probabilities are meaningful not for the single event but only\nas a property of the ensemble under consideration.\n\nThis can also be seen from the mathematical foundations. Probabilities\nare determined by measures on the set of elementary events.\nAll statements in measure theory are _only_ about expectations and\nprobabilities of all possible realizations simultaneously, and say\nnothing at all about any particular realization.\n\nFor a finite binary random sequence with independent bits,\nthe sequence 111111111 has exactly the same status and probability\nas the sequence 101001101 or 000000000, although only the second\nlooks random.\n\n\nArnold Neumaier\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Aaron Denney wrote:
> On [itex]2004-08-12,[/itex] Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
>
>>Probabilities of single events are meaningless.
>
> Oh boy. This is a really blatant misconception of probability.

The misconception seems to be completely on your side.

What is the probability that 'I will die of cancer'?
This is a single event that either will happen, or will not happen.
If you consider this single event only, the probability is 1 or
depending on what will actually happen. (But this sort of probability
is not what we talk about in physics.)

On the other hand one may assign a probability based on some facts
about me. These facts determine an ensemble of people, from which
one can form a statistical estimate of the probability.

It clearly depends on which sort of ensemble one regarde me
to belong to, what probability you will assign.
I belong to many ensembles, and the answer is different for each
of these.

Thus probabilities are meaningful not for the single event but only
as a property of the ensemble under consideration.

This can also be seen from the mathematical foundations. Probabilities
are determined by measures on the set of elementary events.
All statements in measure theory are _only_ about expectations and
probabilities of all possible realizations simultaneously, and say
nothing at all about any particular realization.

For a finite binary random sequence with independent bits,
the sequence 111111111 has exactly the same status and probability
as the sequence 101001101 or 000000000, although only the second
looks random.

Arnold Neumaier

Daryl McCullough

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nArnold Neumaier says...\n\n>What is the probability that \'I will die of cancer\'?\n>This is a single event that either will happen, or will not happen.\n>If you consider this single event only, the probability is 1 or 0\n>depending on what will actually happen. (But this sort of probability\n>is not what we talk about in physics.)\n>\n>On the other hand one may assign a probability based on some facts\n>about me. These facts determine an ensemble of people, from which\n>one can form a statistical estimate of the probability.\n\nI don\'t see how ensembles help give any more precise meaning to\nprobability. Suppose you say that "The probability that someone\nin risk group A will die of cancer is 1/3". That doesn\'t mean\nthat for any 3 people in group A, 1 of them will die of cancer.\nIt doesn\'t mean that for any 30 people, 10 of them will die of\ncancer. It doesn\'t even mean that in the limit as N as goes to\ninfinity, the ratio f_N = #who die of cancer/N = 1/3. What is true\nis that almost surely f_N goes to 1/3 as N goes to infinity, but\nthat "almost surely" is a probabilistic concept, as well. So you\ncan\'t define probabilities completely in terms of ensembles.\n\nSaying that there is a 1/3 chance that I will die of cancer is\nmeaningful without ensembles if you interpret that as a measure\nof my *belief* that I will die of cancer (1 meaning that I\'m\ncertain I will, 0 meaning that I\'m certain I won\'t). Of course,\nthat\'s unsatisfying because we feel that quantum mechanical\nprobabilities are revealing something objective, rather than\nsubjective.\n\nSo I don\'t know what the resolution is, but I don\'t think ensembles\nare on any firmer ground than any other interpretation of probability.\n\n--\nDaryl McCullough\nIthaca, NY\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Arnold Neumaier says...

>What is the probability that 'I will die of cancer'?
>This is a single event that either will happen, or will not happen.
>If you consider this single event only, the probability is 1 or
>depending on what will actually happen. (But this sort of probability
>is not what we talk about in physics.)
>
>On the other hand one may assign a probability based on some facts
>about me. These facts determine an ensemble of people, from which
>one can form a statistical estimate of the probability.

I don't see how ensembles help give any more precise meaning to
probability. Suppose you say that "The probability that someone
in risk group A will die of cancer is 1/3". That doesn't mean
that for any 3 people in group A, 1 of them will die of cancer.
It doesn't mean that for any 30 people, 10 of them will die of
cancer. It doesn't even mean that in the limit as N as goes to
infinity, the ratio [itex]f_N =[/itex] #who die of cancer/N [itex]= 1/3[/itex]. What is true
is that almost surely [itex]f_N[/itex] goes to 1/3 as N goes to infinity, but
that "almost surely" is a probabilistic concept, as well. So you
can't define probabilities completely in terms of ensembles.

Saying that there is [itex]a 1/3[/itex] chance that I will die of cancer is
meaningful without ensembles if you interpret that as a measure
of my *belief* that I will die of cancer (1 meaning that I'm
certain I will, meaning that I'm certain I won't). Of course,
that's unsatisfying because we feel that quantum mechanical
probabilities are revealing something objective, rather than
subjective.

So I don't know what the resolution is, but I don't think ensembles
are on any firmer ground than any other interpretation of probability.

--
Daryl McCullough
Ithaca, NY

Aaron Denney

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\nOn 2004-08-13, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n> Aaron Denney wrote:\n>> On 2004-08-12, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n>>\n>>>Probabilities of single events are meaningless.\n>>\n>> Oh boy. This is a really blatant misconception of probability.\n>\n> The misconception seems to be completely on your side.\n>\n> What is the probability that \'I will die of cancer\'?\n> This is a single event that either will happen, or will not happen.\n\nYep. Events don\'t have to "half-happen" to have a probability of 0.5.\n\n> If you consider this single event only, the probability is 1 or 0\n> depending on what will actually happen.\n\nAfter the fact, yes. Before no, unless you know enough to time\nevolve the system.\n\nThis same argument would also lead to the probability of heads on a coin\nflip being 0 or 1, depending on exactly how one flips it. Actually,\nif you know enough about the flip beforehand, these would be the proper\nones, no matter how fair the coin is.\n\n> (But this sort of probability is not what we talk about in physics.)\n> On the other hand one may assign a probability based on some facts\n> about me.\n\nIt isn\'t? It sounds like we would all the time, and it is just a\nlimiting case of your example.\n\nIf we have all the facts we can push the probabilities to 0 or 1.\n\n(speaking classically. A full QM treatmnet of your life\nwould involve interactions with radioactive particles, which\nwill undoubtedly keep it from being exactly 0 or 1, though\nlaws of large numbers will probably push near 0 or 1 -- either\nyou got enough or you didn\'t.)\n\n> These facts determine an ensemble of people, from which\n> one can form a statistical estimate of the probability.\n\nBut this step is unnecessary.\n\n> It clearly depends on which sort of ensemble one regarde me to belong\n> to, what probability you will assign. I belong to many ensembles, and\n> the answer is different for each of these.\n\nCan you tell me which one is the right one to use for this question?\nWhy or why not? Since you said it is either 0 or 1, which ensemble\ngives that answer?\n\nYou don\'t belong to _any_ ensemble. They\'re sometimes useful for\nconstructing models, but they\'re not the only way to do it.\n\nIf you want to assign something a probability of 1/3, that doesn\'t\nrequire the construction of an ensemble of size 3N, which includes N\nways that event can happen.\n\n> Thus probabilities are meaningful not for the single event but only\n> as a property of the ensemble under consideration.\n\nAnd yet people bet on individual events all the time.\n\n> This can also be seen from the mathematical foundations. Probabilities\n> are determined by measures on the set of elementary events.\n\nThat\'s one way of defining the axioms of probability theory and getting\nthe standard results for manipulating probabilities in various self\nconsistent and maximally useful ways. It\'s not the only way, though\nit can give a nice intuitive understanding (assuming one knows\nmeasure theory better than probability theory, which seems unlikely).\n\nIf you want to invoke measure theory, but the measures are now the\nprobabilities, and the measures are not determined by the ensembles, the\nset you choose for the elementary events.\n\n> All statements in measure theory are _only_ about expectations and\n> probabilities of all possible realizations simultaneously, and say\n> nothing at all about any particular realization.\n\nAll statements in measure theory are about, well, measures over\nsets and subsets. If you\'re modeling probability with it, then\nthe measure over a subset _is_ supposed to be the probability\nof that subset occuring, or the probability of those particular\nrealizations. Of course they don\'t say that the event will or\nwill not happen, unless the probability is zero or one.\n\n> For a finite binary random sequence with independent bits,\n> the sequence 111111111 has exactly the same status and probability\n> as the sequence 101001101 or 000000000, although only the second\n> looks random.\n\nI\'m not sure what you\'re getting at here. Were this phrased\na bit more precisely, I wouldn\'t disagree.\n\nHow do you define "random" and "independent" for each bit, if single\nevents don\'t have probabilities? If 1 is more or less probable than\n0 for each digit, they will indeed have different statuses and\nprobabilities.\n\n--\nAaron Denney\n-><-\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On [itex]2004-08-13,[/itex] Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
> Aaron Denney wrote:
>> On [itex]2004-08-12,[/itex] Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
>>
>>>Probabilities of single events are meaningless.
>>
>> Oh boy. This is a really blatant misconception of probability.
>
> The misconception seems to be completely on your side.
>
> What is the probability that 'I will die of cancer'?
> This is a single event that either will happen, or will not happen.

Yep. Events don't have to "half-happen" to have a probability of .5.

> If you consider this single event only, the probability is 1 or
> depending on what will actually happen.

After the fact, yes. Before no, unless you know enough to time
evolve the system.

This same argument would also lead to the probability of heads on a coin
flip being or 1, depending on exactly how one flips it. Actually,
if you know enough about the flip beforehand, these would be the proper
ones, no matter how fair the coin is.

> (But this sort of probability is not what we talk about in physics.)
> On the other hand one may assign a probability based on some facts
> about me.

It isn't? It sounds like we would all the time, and it is just a
limiting case of your example.

If we have all the facts we can push the probabilities to or 1.

(speaking classically. A full QM treatmnet of your life
would involve interactions with radioactive particles, which
will undoubtedly keep it from being exactly or 1, though
laws of large numbers will probably push near or 1 -- either
you got enough or you didn't.)

> These facts determine an ensemble of people, from which
> one can form a statistical estimate of the probability.

But this step is unnecessary.

> It clearly depends on which sort of ensemble one regarde me to belong
> to, what probability you will assign. I belong to many ensembles, and
> the answer is different for each of these.

Can you tell me which one is the right one to use for this question?
Why or why not? Since you said it is either or 1, which ensemble
gives that answer?

You don't belong to _any_ ensemble. They're sometimes useful for
constructing models, but they're not the only way to do it.

If you want to assign something a probability of [itex]1/3,[/itex] that doesn't
require the construction of an ensemble of size 3N, which includes N
ways that event can happen.

> Thus probabilities are meaningful not for the single event but only
> as a property of the ensemble under consideration.

And yet people bet on individual events all the time.

> This can also be seen from the mathematical foundations. Probabilities
> are determined by measures on the set of elementary events.

That's one way of defining the axioms of probability theory and getting
the standard results for manipulating probabilities in various self
consistent and maximally useful ways. It's not the only way, though
it can give a nice intuitive understanding (assuming one knows
measure theory better than probability theory, which seems unlikely).

If you want to invoke measure theory, but the measures are now the
probabilities, and the measures are not determined by the ensembles, the
set you choose for the elementary events.

> All statements in measure theory are _only_ about expectations and
> probabilities of all possible realizations simultaneously, and say
> nothing at all about any particular realization.

All statements in measure theory are about, well, measures over
sets and subsets. If you're modeling probability with it, then
the measure over a subset _is_ supposed to be the probability
of that subset occuring, or the probability of those particular
realizations. Of course they don't say that the event will or
will not happen, unless the probability is zero or one.

> For a finite binary random sequence with independent bits,
> the sequence 111111111 has exactly the same status and probability
> as the sequence 101001101 or 000000000, although only the second
> looks random.

I'm not sure what you're getting at here. Were this phrased
a bit more precisely, I wouldn't disagree.

How do you define "random" and "independent" for each bit, if single
events don't have probabilities? If 1 is more or less probable than
for each digit, they will indeed have different statuses and
probabilities.

--
Aaron Denney
[itex]-><-[/itex]

Nick Maclaren

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nIn article <411CA50D.8080100@univie.ac.at>,\nArnold Neumaier <Arnold.Neumaier@univie.ac.at> writes:\n|> Aaron Denney wrote:\n|> > On 2004-08-12, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n|> >\n|> >>Probabilities of single events are meaningless.\n|> >\n|> > Oh boy. This is a really blatant misconception of probability.\n|>\n|> The misconception seems to be completely on your side.\n|>\n|> What is the probability that \'I will die of cancer\'?\n|> This is a single event that either will happen, or will not happen.\n|> If you consider this single event only, the probability is 1 or 0\n|> depending on what will actually happen. (But this sort of probability\n|> is not what we talk about in physics.)\n|>\n|> On the other hand one may assign a probability based on some facts\n|> about me. These facts determine an ensemble of people, from which\n|> one can form a statistical estimate of the probability.\n\nI am sorry, but you are wrong on both counts. There are several\nconcepts of probability, and mathematical statisticians work with\nall except the erroneous one:\n\nThe purest mathematical one is a specialisation of measure\ntheory, and probabilities are simply positive measures of sigma\none over some Borel set. Nice and easy, but PURE mathematics.\n\nThe simplest semi-physical approximation is a \'repeatable\'\nexperiment (let\'s leave the philosophy of repeatability out of\nit), which is what most people are taught at a naive level.\n\nThere is a very common error where excessive simplification\nleads people to confuse the concepts of a distribution of a\nmeasurement over a sample and a probability. You have made that\nerror, I am afraid, though not in its simple form.\n\nThere is, however, also the concept of the probability of a\nnon-repeatable event, which can be handled mathematically just\nas easily as a repeatable experiment. What you can\'t do is to\nMEASURE such probabilities, though you can do some measurement\nwith ensembles (as you correctly state).\n\nNote, however, that this all depends on the existence of time\'s\narrow (causality again!), because the probability has a meaning\nonly up until the time that the event takes place (and it may\nchange as time progresses, too). Whereafter, it is either 0 or 1,\ntrue. So, if you are working with a model of the universe that\ndoes not have such a concept, your first statement is correct.\nBut few physicists do.\n\n\nRegards,\nNick Maclaren.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <411CA50D.8080100@univie.ac.at>,
Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes:
|> Aaron Denney wrote:
[itex]|> >[/itex] On [itex]2004-08-12,[/itex] Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
[itex]|> >[/itex]
|> >>Probabilities of single events are meaningless.
[itex]|> >|> > Oh[/itex] boy. This is a really blatant misconception of probability.
|>
|> The misconception seems to be completely on your side.
|>
|> What is the probability that 'I will die of cancer'?
|> This is a single event that either will happen, or will not happen.
|> If you consider this single event only, the probability is 1 or
|> depending on what will actually happen. (But this sort of probability
|> is not what we talk about in physics.)
|>
|> On the other hand one may assign a probability based on some facts
|> about me. These facts determine an ensemble of people, from which
|> one can form a statistical estimate of the probability.

I am sorry, but you are wrong on both counts. There are several
concepts of probability, and mathematical statisticians work with
all except the erroneous one:

The purest mathematical one is a specialisation of measure
theory, and probabilities are simply positive measures of [itex]\sigma[/itex]
one over some Borel set. Nice and easy, but PURE mathematics.

The simplest semi-physical approximation is a 'repeatable'
experiment (let's leave the philosophy of repeatability out of
it), which is what most people are taught at a naive level.

There is a very common error where excessive simplification
leads people to confuse the concepts of a distribution of a
measurement over a sample and a probability. You have made that
error, I am afraid, though not in its simple form.

There is, however, also the concept of the probability of a
non-repeatable event, which can be handled mathematically just
as easily as a repeatable experiment. What you can't do is to
MEASURE such probabilities, though you can do some measurement
with ensembles (as you correctly state).

Note, however, that this all depends on the existence of time's
arrow (causality again!), because the probability has a meaning
only up until the time that the event takes place (and it may
change as time progresses, too). Whereafter, it is either or 1,
true. So, if you are working with a model of the universe that
does not have such a concept, your first statement is correct.
But few physicists do.

Regards,
Nick Maclaren.

Nick Maclaren

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nIn article <slrnchne4b.8n2.wnoise@ofb.net>,\nAaron Denney <wnoise@ofb.net> writes:\n|> On 2004-08-12, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n|> > Probabilities of single events are meaningless.\n|>\n|> Oh boy. This is a really blatant misconception of probability. I can\'t\n|> tell whether the context rescues this (it seems not, though you are\n|> arguing against another \'interpretation\' that I also find odious), but\n|> the words themselves make probability a useless concept.\n\nThat is true :-( But see below for a possible cause of confusion.\n\nIt is also claimed that talking about the probability of an\ninherently non-repeatable event is meaningless, but even that is\nbased on a naive and mistaken view of probability. What is true\nis that it is quite hard to do much with such probabilities.\n\n|> Independence is a statement about non-correlation between probabilities\n|> of single events, which you claim are meaningless.\n\nEr, that is NOT well-phrased! There is also the serious problem\nthat many people may be using "event" to mean what a probabilist\nwould call "outcome". The word "event" is a common and fruitful\nsource of confusion.\n\n\nRegards,\nNick Maclaren.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <slrnchne4b.8n2.wnoise@ofb.net>,
Aaron Denney <wnoise@ofb.net> writes:
|> On [itex]2004-08-12,[/itex] Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
[itex]|> >[/itex] Probabilities of single events are meaningless.
|>
[itex]|> Oh[/itex] boy. This is a really blatant misconception of probability. I can't
|> tell whether the context rescues this (it seems not, though you are
|> arguing against another 'interpretation' that I also find odious), but
|> the words themselves make probability a useless concept.

That is true :-( But see below for a possible cause of confusion.

It is also claimed that talking about the probability of an
inherently non-repeatable event is meaningless, but even that is
based on a naive and mistaken view of probability. What is true
is that it is quite hard to do much with such probabilities.

|> Independence is a statement about non-correlation between probabilities
|> of single events, which you claim are meaningless.

Er, that is NOT well-phrased! There is also the serious problem
that many people may be using "event" to mean what a probabilist
would call "outcome". The word "event" is a common and fruitful
source of confusion.

Regards,
Nick Maclaren.

Arnold Neumaier

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\n\nDaryl McCullough wrote:\n> Arnold Neumaier says...\n>\n>>What is the probability that \'I will die of cancer\'?\n>>This is a single event that either will happen, or will not happen.\n>>If you consider this single event only, the probability is 1 or 0\n>>depending on what will actually happen. (But this sort of probability\n>>is not what we talk about in physics.)\n>>\n>>On the other hand one may assign a probability based on some facts\n>>about me. These facts determine an ensemble of people, from which\n>>one can form a statistical estimate of the probability.\n>\n> I don\'t see how ensembles help give any more precise meaning to\n> probability. Suppose you say that "The probability that someone\n> in risk group A will die of cancer is 1/3". That doesn\'t mean\n> that for any 3 people in group A, 1 of them will die of cancer.\n> It doesn\'t mean that for any 30 people, 10 of them will die of\n> cancer.\n\nI claimed neither of these.\n\nTo say that "The probability that someone in risk group A will die\nof cancer is 1/3" means nothing more or less than that exactly 1/3\nof _all_ people in risk group A will die of cancer.\nOf course, we cannot check this before we have information about\nhow all people in risk group A died, but once we have this information,\nwe know.\n\nUsually we only have incomplete knowledge about the ensemble.\nThis is why statisticians say that they _estimate_ probabilities\nbased on _incomplete_ knowledge, collected from a sample.\nWhereas they _compute_ probabilities from _assumed_complete knowledge\nabout the ensemble, namely the theoretical probability distribution.\nEstimates are usually inaccurate but useful; this reconciles the two\napproaches; hence one finds no difficulties at all in actual practice.\n\nA more extensive discussion can be found in my theoretical physics FAQ at\nhttp://www.mat.univie.ac.at/~neum/physics-faq.txt\nIt currently has the following entries about classical probability:\n5a. Random numbers in probability theory\n5b. How meaningful are probabilities of single events?\n5c. What about the subjective interpretation of probabilities?\n5d. What is the meaning of probabilities?\n5e. How do probabilities apply in practice?\n5f. Priors and entropy in probability theory\n\n\n> Saying that there is a 1/3 chance that I will die of cancer is\n> meaningful without ensembles if you interpret that as a measure\n> of my *belief* that I will die of cancer (1 meaning that I\'m\n> certain I will, 0 meaning that I\'m certain I won\'t). Of course,\n> that\'s unsatisfying because we feel that quantum mechanical\n> probabilities are revealing something objective, rather than\n> subjective.\n\nThis is also unsatisfactory because "belief" is an even poorer defined\nconcept than probability, that depends on psychology of human beings,\nwhich is not a sound foundation for physics.\n\nOn the other hands, "ensemble" is a precise concept with far-reaching\napplications, independent of any beliefs, and hence adequate for use\nin quantum mechanics.\n\n\nArnold Neumaier\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Daryl McCullough wrote:
> Arnold Neumaier says...
>
>>What is the probability that 'I will die of cancer'?
>>This is a single event that either will happen, or will not happen.
>>If you consider this single event only, the probability is 1 or
>>depending on what will actually happen. (But this sort of probability
>>is not what we talk about in physics.)
>>
>>On the other hand one may assign a probability based on some facts
>>about me. These facts determine an ensemble of people, from which
>>one can form a statistical estimate of the probability.
>
> I don't see how ensembles help give any more precise meaning to
> probability. Suppose you say that "The probability that someone
> in risk group A will die of cancer is 1/3". That doesn't mean
> that for any 3 people in group A, 1 of them will die of cancer.
> It doesn't mean that for any 30 people, 10 of them will die of
> cancer.

I claimed neither of these.

To say that "The probability that someone in risk group A will die
of cancer is 1/3" means nothing more or less than that exactly 1/3
of _all_ people in risk group A will die of cancer.
Of course, we cannot check this before we have information about
how all people in risk group A died, but once we have this information,
we know.

Usually we only have incomplete knowledge about the ensemble.
This is why statisticians say that they _estimate_ probabilities
based on _incomplete_ knowledge, collected from a sample.
Whereas they _compute_ probabilities from [itex]_assumed_complete[/itex] knowledge
about the ensemble, namely the theoretical probability distribution.
Estimates are usually inaccurate but useful; this reconciles the two
approaches; hence one finds no difficulties at all in actual practice.

A more extensive discussion can be found in my theoretical physics FAQ at
http://www.mat.univie.ac.at/~neum/physics-faq.txt
It currently has the following entries about classical probability:
5a. Random numbers in probability theory
5b. How meaningful are probabilities of single events?
5c. What about the subjective interpretation of probabilities?
5d. What is the meaning of probabilities?
5e. How do probabilities apply in practice?
5f. Priors and entropy in probability theory

> Saying that there is [itex]a 1/3[/itex] chance that I will die of cancer is
> meaningful without ensembles if you interpret that as a measure
> of my *belief* that I will die of cancer (1 meaning that I'm
> certain I will, meaning that I'm certain I won't). Of course,
> that's unsatisfying because we feel that quantum mechanical
> probabilities are revealing something objective, rather than
> subjective.

This is also unsatisfactory because "belief" is an even poorer defined
concept than probability, that depends on psychology of human beings,
which is not a sound foundation for physics.

On the other hands, "ensemble" is a precise concept with far-reaching
applications, independent of any beliefs, and hence adequate for use
in quantum mechanics.

Arnold Neumaier

Arnold Neumaier

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\n\nNick Maclaren wrote:\n> In article <411CA50D.8080100@univie.ac.at>,\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes:\n> |>\n> |> What is the probability that \'I will die of cancer\'?\n> |> This is a single event that either will happen, or will not happen.\n> |> If you consider this single event only, the probability is 1 or 0\n> |> depending on what will actually happen. (But this sort of probability\n> |> is not what we talk about in physics.)\n> |>\n> |> On the other hand one may assign a probability based on some facts\n> |> about me. These facts determine an ensemble of people, from which\n> |> one can form a statistical estimate of the probability.\n>\n> I am sorry, but you are wrong on both counts. There are several\n> concepts of probability, and mathematical statisticians work with\n> all except the erroneous one:\n>\n> The purest mathematical one is a specialisation of measure\n> theory, and probabilities are simply positive measures of sigma\n> one over some Borel set. Nice and easy, but PURE mathematics.\n\nThis case conforms to my statements. A discrete measure with\nprobabilities that are integral multiples of N can be viewed as\nan ensemble of N experiments. All statements that make sense for\ndiscrete measures make corresponding assertions about such an\nensemble. From a practical point of view, all other measures are\njust useful approximations to summarise the information in\nensembles with very large N.\n\n\n> The simplest semi-physical approximation is a \'repeatable\'\n> experiment (let\'s leave the philosophy of repeatability out of\n> it), which is what most people are taught at a naive level.\n\nHere the concept of probzbility is already dubious, unless you\nspecify which set of repetitions (i.e., the ensemble)\nyou are applying it to.\n\n\n> There is a very common error where excessive simplification\n> leads people to confuse the concepts of a distribution of a\n> measurement over a sample and a probability. You have made that\n> error, I am afraid, though not in its simple form.\n\nNo. There is a sample distribution, and there is a theoretical\ndistribution. Both assign probabilities to events. The sample\ndistribution is the right one if the ensemble is taken as the\nsample you know, the theoretical distribution is the right one\nif the ensemble is taken as the set of all element in the set\nupon which the sigma algebra is based.\n\n\n> There is, however, also the concept of the probability of a\n> non-repeatable event, which can be handled mathematically just\n> as easily as a repeatable experiment.\n\nIf it is as easy, please give the mathematical formulation of the\nprobability that the earth will be hit by an asteroid in the year 9999?\n\n\n> What you can\'t do is to\n> MEASURE such probabilities, though you can do some measurement\n> with ensembles (as you correctly state).\n\nAs I mentioned in another post, probability assignments to single events\ncan be neither verified nor falsified. Thus they are meaningless.\n\n\nArnold Neumaier\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Nick Maclaren wrote:
> In article <411CA50D.8080100@univie.ac.at>,
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes:
> |>
> |> What is the probability that 'I will die of cancer'?
> |> This is a single event that either will happen, or will not happen.
> |> If you consider this single event only, the probability is 1 or
> |> depending on what will actually happen. (But this sort of probability
> |> is not what we talk about in physics.)
> |>
> |> On the other hand one may assign a probability based on some facts
> |> about me. These facts determine an ensemble of people, from which
> |> one can form a statistical estimate of the probability.
>
> I am sorry, but you are wrong on both counts. There are several
> concepts of probability, and mathematical statisticians work with
> all except the erroneous one:
>
> The purest mathematical one is a specialisation of measure
> theory, and probabilities are simply positive measures of [itex]\sigma[/itex]
> one over some Borel set. Nice and easy, but PURE mathematics.

This case conforms to my statements. A discrete measure with
probabilities that are integral multiples of N can be viewed as
an ensemble of N experiments. All statements that make sense for
discrete measures make corresponding assertions about such an
ensemble. From a practical point of view, all other measures are
just useful approximations to summarise the information in
ensembles with very large N.

> The simplest semi-physical approximation is a 'repeatable'
> experiment (let's leave the philosophy of repeatability out of
> it), which is what most people are taught at a naive level.

Here the concept of probzbility is already dubious, unless you
specify which set of repetitions (i.e., the ensemble)
you are applying it to.

> There is a very common error where excessive simplification
> leads people to confuse the concepts of a distribution of a
> measurement over a sample and a probability. You have made that
> error, I am afraid, though not in its simple form.

No. There is a sample distribution, and there is a theoretical
distribution. Both assign probabilities to events. The sample
distribution is the right one if the ensemble is taken as the
sample you know, the theoretical distribution is the right one
if the ensemble is taken as the set of all element in the set
upon which the [itex]\sigma[/itex] algebra is based.

> There is, however, also the concept of the probability of a
> non-repeatable event, which can be handled mathematically just
> as easily as a repeatable experiment.

If it is as easy, please give the mathematical formulation of the
probability that the earth will be hit by an asteroid in the year 9999?

> What you can't do is to
> MEASURE such probabilities, though you can do some measurement
> with ensembles (as you correctly state).

As I mentioned in another post, probability assignments to single events
can be neither verified nor falsified. Thus they are meaningless.

Arnold Neumaier

Arnold Neumaier

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\n\nAaron Denney wrote:\n> On 2004-08-13, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n>\n>>What is the probability that \'I will die of cancer\'?\n>>This is a single event that either will happen, or will not happen.\n>\n> Yep. Events don\'t have to "half-happen" to have a probability of 0.5.\n\nProbability assignments to single events can be neither verified nor\nfalsified. Indeed, suppose we intend to throw a coin exactly once.\nPerson A claims \'the probability of the coin coming out head is 50%\'.\nPerson B claims \'the probability of the coin coming out head is 20%\'.\nPerson C claims \'the probability of the coin coming out head is 80%\'.\nNow we throw the coin and find \'head\'. Who was right? It is undecidable.\n\nThus there cannot be objective content in the statement\n\'the probability of the coin coming out head is p\',\nwhen applied to a single case. Subjectively, of course, every person\nmay feel (and is entitled to feel) right about their probability assignment.\nBut for use in science, such a subjective view (where everyone is right,\nno matter which statement was made) is completely useless.\n\n\n\n>>If you consider this single event only, the probability is 1 or 0\n>>depending on what will actually happen.\n>\n> After the fact, yes. Before no, unless you know enough to time\n> evolve the system.\n\nWhat if someone knows the fact and someone else doesn\'t???\nI am discussing objective probability, since physics is an objective\nscience.\n\n\n\n>>It clearly depends on which sort of ensemble one regarde me to belong\n>>to, what probability you will assign. I belong to many ensembles, and\n>>the answer is different for each of these.\n>\n> Can you tell me which one is the right one to use for this question?\n> Why or why not? Since you said it is either 0 or 1, which ensemble\n> gives that answer?\n\nThe ensemble consisting of me only, as appropriate for a single case.\n\nIn other ensembles, the probability is just the proportion of\npeople in the ensemble dying of cancer; of course this probability,\nthough it is a well-defined number, can be estimated only approximately\n- at least until I am dead ;-)\n\n\n>>Thus probabilities are meaningful not for the single event but only\n>>as a property of the ensemble under consideration.\n>\n> And yet people bet on individual events all the time.\n\nOh yes. They estimate probabilities, based on their faforite ensemble.\nBut as you know, people often lose their bets!\n\n\n>>This can also be seen from the mathematical foundations. Probabilities\n>>are determined by measures on the set of elementary events.\n>\n> That\'s one way of defining the axioms of probability theory and getting\n> the standard results for manipulating probabilities in various self\n> consistent and maximally useful ways. It\'s not the only way,\n\nBut it is the only consistent way.\n\n\n>>All statements in measure theory are _only_ about expectations and\n>>probabilities of all possible realizations simultaneously, and say\n>>nothing at all about any particular realization.\n>\n> All statements in measure theory are about, well, measures over\n> sets and subsets. If you\'re modeling probability with it, then\n> the measure over a subset _is_ supposed to be the probability\n\n"_is_", not "_is_ supposed to be".\n\n> of that subset occuring, or the probability of those particular\n> realizations. Of course they don\'t say that the event will or\n> will not happen, unless the probability is zero or one.\n\nYes; this is why they say nothing at all about the single case.\n\n\n>>For a finite binary random sequence with independent bits,\n>>the sequence 111111111 has exactly the same status and probability\n>>as the sequence 101001101 or 000000000, although only the second\n>>looks random.\n>\n> I\'m not sure what you\'re getting at here. Were this phrased\n> a bit more precisely, I wouldn\'t disagree.\n>\n> How do you define "random" and "independent" for each bit, if single\n> events don\'t have probabilities? If 1 is more or less probable than\n> 0 for each digit, they will indeed have different statuses and\n> probabilities.\n\nHere I meant \'random bit\' to say both probabilities 1/2.\nA random sequence is _not_ a sequence of numbers but a sequence of\nrandom numbers. Only the realizations are sequences of ordinary\nnumbers. Sequences of ordinary numbers are _never_ random, but they\ncan \'look random\' (in a subjective sense).\n\n\nArnold Neumaier\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Aaron Denney wrote:
> On [itex]2004-08-13,[/itex] Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
>
>>What is the probability that 'I will die of cancer'?
>>This is a single event that either will happen, or will not happen.
>
> Yep. Events don't have to "half-happen" to have a probability of .5.

Probability assignments to single events can be neither verified nor
falsified. Indeed, suppose we intend to throw a coin exactly once.
Person A claims 'the probability of the coin coming out head is 50%'.
Person B claims 'the probability of the coin coming out head is 20%'.
Person C claims 'the probability of the coin coming out head is 80%'.
Now we throw the coin and find 'head'. Who was right? It is undecidable.

Thus there cannot be objective content in the statement
'the probability of the coin coming out head is p',
when applied to a single case. Subjectively, of course, every person
may feel (and is entitled to feel) right about their probability assignment.
But for use in science, such a subjective view (where everyone is right,
no matter which statement was made) is completely useless.

>>If you consider this single event only, the probability is 1 or
>>depending on what will actually happen.
>
> After the fact, yes. Before no, unless you know enough to time
> evolve the system.

What if someone knows the fact and someone else doesn't???
I am discussing objective probability, since physics is an objective
science.

>>It clearly depends on which sort of ensemble one regarde me to belong
>>to, what probability you will assign. I belong to many ensembles, and
>>the answer is different for each of these.
>
> Can you tell me which one is the right one to use for this question?
> Why or why not? Since you said it is either or 1, which ensemble
> gives that answer?

The ensemble consisting of me only, as appropriate for a single case.

In other ensembles, the probability is just the proportion of
people in the ensemble dying of cancer; of course this probability,
though it is a well-defined number, can be estimated only approximately
[itex]- at[/itex] least until I am dead ;-)

>>Thus probabilities are meaningful not for the single event but only
>>as a property of the ensemble under consideration.
>
> And yet people bet on individual events all the time.

Oh yes. They estimate probabilities, based on their faforite ensemble.
But as you know, people often lose their bets!

>>This can also be seen from the mathematical foundations. Probabilities
>>are determined by measures on the set of elementary events.
>
> That's one way of defining the axioms of probability theory and getting
> the standard results for manipulating probabilities in various self
> consistent and maximally useful ways. It's not the only way,

But it is the only consistent way.

>>All statements in measure theory are _only_ about expectations and
>>probabilities of all possible realizations simultaneously, and say
>>nothing at all about any particular realization.
>
> All statements in measure theory are about, well, measures over
> sets and subsets. If you're modeling probability with it, then
> the measure over a subset _is_ supposed to be the probability

"_is_", not "_is_ supposed to be".

> of that subset occuring, or the probability of those particular
> realizations. Of course they don't say that the event will or
> will not happen, unless the probability is zero or one.

Yes; this is why they say nothing at all about the single case.

>>For a finite binary random sequence with independent bits,
>>the sequence 111111111 has exactly the same status and probability
>>as the sequence 101001101 or 000000000, although only the second
>>looks random.
>
> I'm not sure what you're getting at here. Were this phrased
> a bit more precisely, I wouldn't disagree.
>
> How do you define "random" and "independent" for each bit, if single
> events don't have probabilities? If 1 is more or less probable than
> for each digit, they will indeed have different statuses and
> probabilities.

Here I meant 'random bit' to say both probabilities 1/2.
A random sequence is [itex]_not_ a[/itex] sequence of numbers but a sequence of
random numbers. Only the realizations are sequences of ordinary
numbers. Sequences of ordinary numbers are _never_ random, but they
can 'look random' (in a subjective sense).

Arnold Neumaier

Nick Maclaren

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nIn article <411F45F1.10704@univie.ac.at>,\nArnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n>Daryl McCullough wrote:\n>\n>To say that "The probability that someone in risk group A will die\n>of cancer is 1/3" means nothing more or less than that exactly 1/3\n>of _all_ people in risk group A will die of cancer.\n\nThat is completely and utterly wrong. You can see that by taking\na nice, simple example (i.e. not cancer).\n\nFair coins have a probability 0.5 of coming up heads. If you toss\n10 fair coins, it is NOT necessarily the case that exactly 5 will\nshow heads.\n\nYou get exactly the same situation with a fixed number of electrons\nin quantum mechanics.\n\n>Of course, we cannot check this before we have information about\n>how all people in risk group A died, but once we have this information,\n>we know.\n\nLet us say 7 coins out of 10 show heads. Would that mean that the\nprobability of a fair coin showing heads was 70%? Oh, come now.\nIf you toss them again (which is where repeatable experiments come\nin), you might well have 4 come up heads. And so on.\n\n>Usually we only have incomplete knowledge about the ensemble.\n>This is why statisticians say that they _estimate_ probabilities\n>based on _incomplete_ knowledge, collected from a sample.\n\nThat is true, but it is only ONE of the things that statisticians\ndo. And not the most important one, either.\n\n>Whereas they _compute_ probabilities from _assumed_complete knowledge\n>about the ensemble, namely the theoretical probability distribution.\n\nGrrk. We also compute confidence intervals on probabilities based\non data, compute probabilities based on a known mathematical model\nand values of its parameters, compute various forms of best estimates\nof probabilities based on data, and so on.\n\n>Estimates are usually inaccurate but useful; this reconciles the two\n>approaches; hence one finds no difficulties at all in actual practice.\n\nHmm. I am glad that you find the problems so simple. Not all leading\nstatisticians do.\n\n>> Saying that there is a 1/3 chance that I will die of cancer is\n>> meaningful without ensembles if you interpret that as a measure\n>> of my *belief* that I will die of cancer (1 meaning that I\'m\n>> certain I will, 0 meaning that I\'m certain I won\'t). Of course,\n>> that\'s unsatisfying because we feel that quantum mechanical\n>> probabilities are revealing something objective, rather than\n>> subjective.\n>\n>This is also unsatisfactory because "belief" is an even poorer defined\n>concept than probability, that depends on psychology of human beings,\n>which is not a sound foundation for physics.\n\nYes and no. That is not true if he defines the mathematical model\nhe is using and the data and methods he is using to estimate the\nparameters of that model. That is, after all, precisely the\nstatistical analogue of measuring a physical constant!\n\n>On the other hands, "ensemble" is a precise concept with far-reaching\n>applications, independent of any beliefs, and hence adequate for use\n>in quantum mechanics.\n\nWell, it wasn\'t a standard term when I did my (masters equivalent)\ncourse in mathematical statistics, though that was some 30+ years\nback. I can guess what it means, but I don\'t think that "a precise\nconcept" is what a mathematical statistician would call it.\n\nPlease note that I am replying to this posting and not yours to mine,\nbecause it gives a clearer example of some of the issues.\n\n\nRegards,\nNick Maclaren.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <411F45F1.10704@univie.ac.at>,
Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
>Daryl McCullough wrote:
>
>To say that "The probability that someone in risk group A will die
>of cancer is 1/3" means nothing more or less than that exactly 1/3
>of _all_ people in risk group A will die of cancer.

That is completely and utterly wrong. You can see that by taking
a nice, simple example (i.e. not cancer).

Fair coins have a probability .5 of coming up heads. If you toss
10 fair coins, it is NOT necessarily the case that exactly 5 will
show heads.

You get exactly the same situation with a fixed number of electrons
in quantum mechanics.

>Of course, we cannot check this before we have information about
>how all people in risk group A died, but once we have this information,
>we know.

Let us say 7 coins out of 10 show heads. Would that mean that the
probability of a fair coin showing heads was 70%? Oh, come now.
If you toss them again (which is where repeatable experiments come
in), you might well have 4 come up heads. And so on.

>Usually we only have incomplete knowledge about the ensemble.
>This is why statisticians say that they _estimate_ probabilities
>based on _incomplete_ knowledge, collected from a sample.

That is true, but it is only ONE of the things that statisticians
do. And not the most important one, either.

>Whereas they _compute_ probabilities from [itex]_assumed_complete[/itex] knowledge
>about the ensemble, namely the theoretical probability distribution.

Grrk. We also compute confidence intervals on probabilities based
on data, compute probabilities based on a known mathematical model
and values of its parameters, compute various forms of best estimates
of probabilities based on data, and so on.

>Estimates are usually inaccurate but useful; this reconciles the two
>approaches; hence one finds no difficulties at all in actual practice.

Hmm. I am glad that you find the problems so simple. Not all leading
statisticians do.

>> Saying that there is [itex]a 1/3[/itex] chance that I will die of cancer is
>> meaningful without ensembles if you interpret that as a measure
>> of my *belief* that I will die of cancer (1 meaning that I'm
>> certain I will, meaning that I'm certain I won't). Of course,
>> that's unsatisfying because we feel that quantum mechanical
>> probabilities are revealing something objective, rather than
>> subjective.
>
>This is also unsatisfactory because "belief" is an even poorer defined
>concept than probability, that depends on psychology of human beings,
>which is not a sound foundation for physics.

Yes and no. That is not true if he defines the mathematical model
he is using and the data and methods he is using to estimate the
parameters of that model. That is, after all, precisely the
statistical analogue of measuring a physical constant!

>On the other hands, "ensemble" is a precise concept with far-reaching
>applications, independent of any beliefs, and hence adequate for use
>in quantum mechanics.

Well, it wasn't a standard term when I did my (masters equivalent)
course in mathematical statistics, though that was some 30+ years
back. I can guess what it means, but I don't think that "a precise
concept" is what a mathematical statistician would call it.

Please note that I am replying to this posting and not yours to mine,
because it gives a clearer example of some of the issues.

Regards,
Nick Maclaren.

Arnold Neumaier

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\n\nNick Maclaren wrote:\n> In article <411F45F1.10704@univie.ac.at>,\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n>\n>>Daryl McCullough wrote:\n>>\n>>To say that "The probability that someone in risk group A will die\n>>of cancer is 1/3" means nothing more or less than that exactly 1/3\n>>of _all_ people in risk group A will die of cancer.\n>\n>\n> That is completely and utterly wrong. You can see that by taking\n> a nice, simple example (i.e. not cancer).\n>\n> Fair coins have a probability 0.5 of coming up heads. If you toss\n> 10 fair coins, it is NOT necessarily the case that exactly 5 will\n> show heads.\n\nThis is not a correct translation of my claim.\n\nIf you take any finite sigma algebra representing a\nfair coin, one has a finite ensemble of elementary events,\nand exactly half of them come out heads. If you take an infinite\nsigma algebra, the ensemble is infinite, but with the natural\nweighting, again exactly half of them come out head.\nThis is precisely what I claimed.\n\n\'Tossing 10 fair coins\' is just a sloppy way of saying\n\'Selecting a sample of size 10 from the total ensemble\',\nand it is obvious that here the number of heads is 5 only on\naverage over many random samples, again as I had claimed in an\nunquoted part of the post to which you replied.\n\n\n>>On the other hands, "ensemble" is a precise concept with far-reaching\n>>applications, independent of any beliefs, and hence adequate for use\n>>in quantum mechanics.\n>\n> Well, it wasn\'t a standard term when I did my (masters equivalent)\n> course in mathematical statistics, though that was some 30+ years\n> back. I can guess what it means, but I don\'t think that "a precise\n> concept" is what a mathematical statistician would call it.\n\nI am talking here (s.p.r.) physics language.\nIn mathematical terms, a classical ensemble is the set of elementary\nevents underlying the sigma algebra over which the measure is defined.\n\n\nArnold Neumaier\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Nick Maclaren wrote:
> In article <411F45F1.10704@univie.ac.at>,
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
>
>>Daryl McCullough wrote:
>>
>>To say that "The probability that someone in risk group A will die
>>of cancer is 1/3" means nothing more or less than that exactly 1/3
>>of _all_ people in risk group A will die of cancer.
>
>
> That is completely and utterly wrong. You can see that by taking
> a nice, simple example (i.e. not cancer).
>
> Fair coins have a probability .5 of coming up heads. If you toss
> 10 fair coins, it is NOT necessarily the case that exactly 5 will
> show heads.

This is not a correct translation of my claim.

If you take any finite [itex]\sigma[/itex] algebra representing a
fair coin, one has a finite ensemble of elementary events,
and exactly half of them come out heads. If you take an infinite
[itex]\sigma[/itex] algebra, the ensemble is infinite, but with the natural
weighting, again exactly half of them come out head.
This is precisely what I claimed.

'Tossing 10 fair coins' is just a sloppy way of saying
'Selecting a sample of size 10 from the total ensemble',
and it is obvious that here the number of heads is 5 only on
average over many random samples, again as I had claimed in an
unquoted part of the post to which you replied.

>>On the other hands, "ensemble" is a precise concept with far-reaching
>>applications, independent of any beliefs, and hence adequate for use
>>in quantum mechanics.
>
> Well, it wasn't a standard term when I did my (masters equivalent)
> course in mathematical statistics, though that was some 30+ years
> back. I can guess what it means, but I don't think that "a precise
> concept" is what a mathematical statistician would call it.

I am talking here (s.p.r.) physics language.
In mathematical terms, a classical ensemble is the set of elementary
events underlying the [itex]\sigma[/itex] algebra over which the measure is defined.

Arnold Neumaier

Daryl McCullough

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nArnold Neumaier says...\n\n>Probability assignments to single events can be neither verified nor\n>falsified.\n\nProbabilistic predictions can *never* be verified or falsified by any\n(finite) number of observations. If the prediction is that half of all\nparticles of type X decay within T seconds, how many measurements does\nit take to prove the prediction is true? How many measurements does\nit take to prove the prediction is false? The answer is that there is\nno number that is sufficient. It\'s just that as we make more and more\nobservations, we become and more confident that the prediction is true\n(or that it\'s false). There is never a point where it is absolutely\nverified or absolutely falsified, although we can get to a point where\nwe are as good as certain one way or the other.\n\nBut for any finite number of observations, we don\'t know whether the\nprobabilistic prediction is true or not. We\'re in the same boat whether\nwe are talking about 1 observation or 1000.\n\n--\nDaryl McCullough\nIthaca, NY\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Arnold Neumaier says...

>Probability assignments to single events can be neither verified nor
>falsified.

Probabilistic predictions can *never* be verified or falsified by any
(finite) number of observations. If the prediction is that half of all
particles of type X decay within T seconds, how many measurements does
it take to prove the prediction is true? How many measurements does
it take to prove the prediction is false? The answer is that there is
no number that is sufficient. It's just that as we make more and more
observations, we become and more confident that the prediction is true
(or that it's false). There is never a point where it is absolutely
verified or absolutely falsified, although we can get to a point where
we are as good as certain one way or the other.

But for any finite number of observations, we don't know whether the
probabilistic prediction is true or not. We're in the same boat whether
we are talking about 1 observation or 1000.

--
Daryl McCullough
Ithaca, NY

Daryl McCullough

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nArnold Neumaier says...\n\n>To say that "The probability that someone in risk group A will die\n>of cancer is 1/3" means nothing more or less than that exactly 1/3\n>of _all_ people in risk group A will die of cancer.\n\nThat is not true. That\'s the *frequency* with which people in\ngroup A die of cancer. It is *not* the probability. The frequency\nis supposed to *approach* the probability in some sense, but they\naren\'t the same thing.\n\nThink about it. If you have a coin with a 50% chance of heads\nand 50% chance of tails, then the frequency jumps around as time\ngoes on. With the first coin toss, the frequency of heads\nis either 0 or 1. With the second coin toss, the frequency is\neither 0, 1/2, or 1. With the third toss, the frequency is either\n0, 1/3, 2/3, or 1.\n\nNobody would say that the *probability* jumps around like that. The\nprobability is always the same (well, unless there is some\ntime-dependent effect).\n\nProbability is *not* the same thing as relative frequency, and\nensembles don\'t help to define probability. They can help define\nrelative frequency, but only in the case of *finite* ensembles\n(frequency is not well-defined for an infinite ensemble). But it\nis exactly in the case of finite ensembles that the difference\nbetween probability and relativity frequency is the most pronounced.\n\n--\nDaryl McCullough\nIthaca, NY\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Arnold Neumaier says...

>To say that "The probability that someone in risk group A will die
>of cancer is 1/3" means nothing more or less than that exactly 1/3
>of _all_ people in risk group A will die of cancer.

That is not true. That's the *frequency* with which people in
group A die of cancer. It is *not* the probability. The frequency
is supposed to *approach* the probability in some sense, but they
aren't the same thing.

Think about it. If you have a coin with a 50% chance of heads
and 50% chance of tails, then the frequency jumps around as time
goes on. With the first coin toss, the frequency of heads
is either or 1. With the second coin toss, the frequency is
either [itex]0, 1/2,[/itex] or 1. With the third toss, the frequency is either
[itex]0, 1/3, 2/3,[/itex] or 1.

Nobody would say that the *probability* jumps around like that. The
probability is always the same (well, unless there is some
time-dependent effect).

Probability is *not* the same thing as relative frequency, and
ensembles don't help to define probability. They can help define
relative frequency, but only in the case of *finite* ensembles
(frequency is not well-defined for an infinite ensemble). But it
is exactly in the case of finite ensembles that the difference
between probability and relativity frequency is the most pronounced.

--
Daryl McCullough
Ithaca, NY

Aaron Denney

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\nOn 2004-08-16, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n> To say that "The probability that someone in risk group A will die\n> of cancer is 1/3" means nothing more or less than that exactly 1/3\n> of _all_ people in risk group A will die of cancer.\n> Of course, we cannot check this before we have information about\n> how all people in risk group A died, but once we have this information,\n> we know.\n\nOkay, you have two choices for defining this ensemble.\nEither (a) it is actual people in this ensemble, and it\'s a finite\nset, or (b) it is imaginary people similar to the ones you actually\ncare about.\n\nFor (b), the probability is not objective, not checkable by anyone else.\n\nFor (a), well, suppose I flip a coin ten times, and get 6 heads and 4\ntails. I really, really, hope that you don\'t think that the coin\nhas a probability of exactly 0.6 of coming up heads. If you flip it\nagain twice and get two tails, is the probability for heads now 0.5?\n\n--\nAaron Denney\n-><-\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On [itex]2004-08-16,[/itex] Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
> To say that "The probability that someone in risk group A will die
> of cancer is 1/3" means nothing more or less than that exactly 1/3
> of _all_ people in risk group A will die of cancer.
> Of course, we cannot check this before we have information about
> how all people in risk group A died, but once we have this information,
> we know.

Okay, you have two choices for defining this ensemble.
Either (a) it is actual people in this ensemble, and it's a finite
set, or (b) it is imaginary people similar to the ones you actually
care about.

For (b), the probability is not objective, not checkable by anyone else.

For (a), well, suppose I flip a coin ten times, and get 6 heads and 4
tails. I really, really, hope that you don't think that the coin
has a probability of exactly .6 of coming up heads. If you flip it
again twice and get two tails, is the probability for heads now .5?

--
Aaron Denney
[itex]-><-[/itex]

Aaron Denney

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\nOn 2004-08-16, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n>\n>\n>\n>\n> Aaron Denney wrote:\n>> On 2004-08-13, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n>>\n>>>What is the probability that \'I will die of cancer\'?\n>>>This is a single event that either will happen, or will not happen.\n>>\n>> Yep. Events don\'t have to "half-happen" to have a probability of 0.5.\n>\n> Probability assignments to single events can be neither verified nor\n> falsified.\n\nRight. Probability assignments inherently have some subjectivity --\nwhat someone knows determines the assignment.\n\n> Indeed, suppose we intend to throw a coin exactly once.\n> Person A claims \'the probability of the coin coming out head is 50%\'.\n> Person B claims \'the probability of the coin coming out head is 20%\'.\n> Person C claims \'the probability of the coin coming out head is 80%\'.\n> Now we throw the coin and find \'head\'. Who was right? It is undecidable.\n\nAny of them, or none of them, depending on what they knew about the\nprior conditions of tossing. "Appropriate" probability assignment would\nbe better language than "correct". All of them could be correct if A\nknows only that it has heads and tails, and that both can come up, if B\nknows that the coin is heavier on the heads side by a certain amount,\nand C knows that the tosser is extremely practiced and can make it come\nout heads 80% of time.\n\n> Thus there cannot be objective content in the statement\n> \'the probability of the coin coming out head is p\',\n> when applied to a single case. Subjectively, of course, every person\n> may feel (and is entitled to feel) right about their probability assignment.\n> But for use in science, such a subjective view (where everyone is right,\n> no matter which statement was made) is completely useless.\n\nNot at all. Suppose someone you trust completely assures you that a\ncoin is biased so that during tests it comes up one way 80% of the\ntime, and the other 20% of the time, but refuses to tell you which\nis which. What is your probability assignment that the coin comes up\nheads on one toss?\n\nI claim this is the same case as flipping any coin. Deterministically\nit will come up whatever it comes up as. P = 0, or 1, if you knew\neverything. Still, even in this case, where the "objective" probability\nis 0.8 or 0.2, the best representation of the information available to\nyou for a single toss is 0.5 to heads and 0.5 to tails.\n\nIf you know it will be flipped twice, you should assign 0.32 to HH and\nTT and 0.18 to HT and TH.\n\n>>>If you consider this single event only, the probability is 1 or 0\n>>>depending on what will actually happen.\n>>\n>> After the fact, yes. Before no, unless you know enough to time\n>> evolve the system.\n>\n> What if someone knows the fact and someone else doesn\'t???\n> I am discussing objective probability, since physics is an objective\n> science.\n\nThen someone gets a better estimate. Probability theory is a\nway of reasoning about uncertainty. If someone knows that they\nknow enough information, they get what will actually happen with\nprobability 1. Someone who doesn\'t know enough will get a whole\nrange of possible outcomes with different probabilities. These\ndifferent probabilities will still be useful information about\nwhat choices to make.\n\nIt\'s okay for the results to be subjective because different people\nstart with different information. If someone starts with incorrect,\nrather than incomplete information, they\'ll get incorrect results.\nThis is expected.\n\nProbability is _not_ figuring out how often something happens in\nrepeatable experiments. It can be applied to that, yielding the\nwell known de Finetti exchangeability results linking long-run\nfrequency with probability, but limiting it to that case is perverse.\n\n>>>It clearly depends on which sort of ensemble one regarde me to belong\n>>>to, what probability you will assign. I belong to many ensembles, and\n>>>the answer is different for each of these.\n>>\n>> Can you tell me which one is the right one to use for this question?\n>> Why or why not? Since you said it is either 0 or 1, which ensemble\n>> gives that answer?\n>\n> The ensemble consisting of me only, as appropriate for a single case.\n>\n> In other ensembles, the probability is just the proportion of\n> people in the ensemble dying of cancer; of course this probability,\n> though it is a well-defined number, can be estimated only approximately\n> - at least until I am dead ;-)\n\nDo you allow infinite ensembles, or are only rational numbers acceptable\nprobabilities?\n\n>>>Thus probabilities are meaningful not for the single event but only\n>>>as a property of the ensemble under consideration.\n>>\n>> And yet people bet on individual events all the time.\n>\n> Oh yes. They estimate probabilities, based on their favorite ensemble.\n> But as you know, people often lose their bets!\n\nSure. That doesn\'t mean they estimated the probability wrong, or are\nmisusing probability theory. Refusing to let probability theory deal\nwith single events reduces its applicability to almost nothing, and\npeople do successfully use it for single events.\n\n>>>This can also be seen from the mathematical foundations. Probabilities\n>>>are determined by measures on the set of elementary events.\n>>\n>> That\'s one way of defining the axioms of probability theory and getting\n>> the standard results for manipulating probabilities in various self\n>> consistent and maximally useful ways. It\'s not the only way,\n>\n> But it is the only consistent way.\n\nThere are at least three or four different ways. They all give the same\nanswers in the areas where they all apply.\n\n>> of that subset occuring, or the probability of those particular\n>> realizations. Of course they don\'t say that the event will or\n>> will not happen, unless the probability is zero or one.\n>\n> Yes; this is why they say nothing at all about the single case.\n\nSure they, just not something definite. That\'s why there probabilities\ninstead of certainties.\n\nNow, you can do most of this reasoning about uncertainty with ensembles\nrather than states of knowledge, but it\'s much harder and more\ncomplicated. You have to make sure that the ensembles you come up with\nare not only consistent with your state of knowledge, but also don\'t\ntell you anything more -- that they aren\'t biased.\n\nThe choice of ensemble to use is just as subjective as the choices of A,\nB, and C in your example above.\n\n--\nAaron Denney\n-><-\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On [itex]2004-08-16,[/itex] Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
>
>
>
>
> Aaron Denney wrote:
>> On [itex]2004-08-13,[/itex] Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
>>
>>>What is the probability that 'I will die of cancer'?
>>>This is a single event that either will happen, or will not happen.
>>
>> Yep. Events don't have to "half-happen" to have a probability of .5.
>
> Probability assignments to single events can be neither verified nor
> falsified.

Right. Probability assignments inherently have some subjectivity --
what someone knows determines the assignment.

> Indeed, suppose we intend to throw a coin exactly once.
> Person A claims 'the probability of the coin coming out head is 50%'.
> Person B claims 'the probability of the coin coming out head is 20%'.
> Person C claims 'the probability of the coin coming out head is 80%'.
> Now we throw the coin and find 'head'. Who was right? It is undecidable.

Any of them, or none of them, depending on what they knew about the
prior conditions of tossing. "Appropriate" probability assignment would
be better language than "correct". All of them could be correct if A
knows only that it has heads and tails, and that both can come up, if B
knows that the coin is heavier on the heads side by a certain amount,
and C knows that the tosser is extremely practiced and can make it come
out heads 80% of time.

> Thus there cannot be objective content in the statement
> 'the probability of the coin coming out head is p',
> when applied to a single case. Subjectively, of course, every person
> may feel (and is entitled to feel) right about their probability assignment.
> But for use in science, such a subjective view (where everyone is right,
> no matter which statement was made) is completely useless.

Not at all. Suppose someone you trust completely assures you that a
coin is biased so that during tests it comes up one way 80% of the
time, and the other 20% of the time, but refuses to tell you which
is which. What is your probability assignment that the coin comes up
heads on one toss?

I claim this is the same case as flipping any coin. Deterministically
it will come up whatever it comes up as. [itex]P = 0,[/itex] or 1, if you knew
everything. Still, even in this case, where the "objective" probability
is .8 or .2, the best representation of the information available to
you for a single toss is .5 to heads and .5 to tails.

If you know it will be flipped twice, you should assign .32 to HH and
TT and .18 to HT and TH.

>>>If you consider this single event only, the probability is 1 or
>>>depending on what will actually happen.
>>
>> After the fact, yes. Before no, unless you know enough to time
>> evolve the system.
>
> What if someone knows the fact and someone else doesn't???
> I am discussing objective probability, since physics is an objective
> science.

Then someone gets a better estimate. Probability theory is a
way of reasoning about uncertainty. If someone knows that they
know enough information, they get what will actually happen with
probability 1. Someone who doesn't know enough will get a whole
range of possible outcomes with different probabilities. These
different probabilities will still be useful information about
what choices to make.

It's okay for the results to be subjective because different people
start with different information. If someone starts with incorrect,
rather than incomplete information, they'll get incorrect results.
This is expected.

Probability is _not_ figuring out how often something happens in
repeatable experiments. It can be applied to that, yielding the
well known de Finetti exchangeability results linking long-run
frequency with probability, but limiting it to that case is perverse.

>>>It clearly depends on which sort of ensemble one regarde me to belong
>>>to, what probability you will assign. I belong to many ensembles, and
>>>the answer is different for each of these.
>>
>> Can you tell me which one is the right one to use for this question?
>> Why or why not? Since you said it is either or 1, which ensemble
>> gives that answer?
>
> The ensemble consisting of me only, as appropriate for a single case.
>
> In other ensembles, the probability is just the proportion of
> people in the ensemble dying of cancer; of course this probability,
> though it is a well-defined number, can be estimated only approximately
> [itex]- at[/itex] least until I am dead ;-)

Do you allow infinite ensembles, or are only rational numbers acceptable
probabilities?

>>>Thus probabilities are meaningful not for the single event but only
>>>as a property of the ensemble under consideration.
>>
>> And yet people bet on individual events all the time.
>
> Oh yes. They estimate probabilities, based on their favorite ensemble.
> But as you know, people often lose their bets!

Sure. That doesn't mean they estimated the probability wrong, or are
misusing probability theory. Refusing to let probability theory deal
with single events reduces its applicability to almost nothing, and
people do successfully use it for single events.

>>>This can also be seen from the mathematical foundations. Probabilities
>>>are determined by measures on the set of elementary events.
>>
>> That's one way of defining the axioms of probability theory and getting
>> the standard results for manipulating probabilities in various self
>> consistent and maximally useful ways. It's not the only way,
>
> But it is the only consistent way.

There are at least three or four different ways. They all give the same
answers in the areas where they all apply.

>> of that subset occuring, or the probability of those particular
>> realizations. Of course they don't say that the event will or
>> will not happen, unless the probability is zero or one.
>
> Yes; this is why they say nothing at all about the single case.

Sure they, just not something definite. That's why there probabilities
instead of certainties.

Now, you can do most of this reasoning about uncertainty with ensembles
rather than states of knowledge, but it's much harder and more
complicated. You have to make sure that the ensembles you come up with
are not only consistent with your state of knowledge, but also don't
tell you anything more -- that they aren't biased.

The choice of ensemble to use is just as subjective as the choices of A,
B, and C in your example above.

--
Aaron Denney
[itex]-><-[/itex]

Nick Maclaren

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nIn article <4121F056.7090303@univie.ac.at>,\nArnold Neumaier <Arnold.Neumaier@univie.ac.atwrote:\n>>>\n>>>To say that "The probability that someone in risk group A will die\n>>>of cancer is 1/3" means nothing more or less than that exactly 1/3\n>>>of _all_ people in risk group A will die of cancer.\n>>\n>That is completely and utterly wrong. You can see that by taking\n>a nice, simple example (i.e. not cancer).\n>>\n>Fair coins have a probability 0.5 of coming up heads. If you toss\n>10 fair coins, it is NOT necessarily the case that exactly 5 will\n>show heads.\n>\n>This is not a correct translation of my claim.\n\nGood, Because it is completely wrong. It IS a correct example of\nwhat you posted, so I am glad that you didn\'t mean it.\n\n> >If you take any finite sigma algebra representing a\n> >fair coin, one has a finite ensemble of elementary events,\n> >and exactly half of them come out heads.\n>\n> If you take an infinite\n>sigma algebra, the ensemble is infinite, but with the natural\n>weighting, again exactly half of them come out head.\n\nThe very concept of "with the natural weighting, again exactly half\nof them come out head" is misleading to a degree when applied to a\nlimit process (which this is). See below.\n\n>\'Tossing 10 fair coins\' is just a sloppy way of saying\n>\'Selecting a sample of size 10 from the total ensemble\',\n>and it is obvious that here the number of heads is 5 only on\n>average over many random samples, again as I had claimed in an\n>unquoted part of the post to which you replied.\n>\n>>>On the other hands, "ensemble" is a precise concept with far-reaching\n>>>applications, independent of any beliefs, and hence adequate for use\n>>>in quantum mechanics.\n>>\n>Well, it wasn\'t a standard term when I did my (masters equivalent)\n>course in mathematical statistics, though that was some 30+ years\n>back. I can guess what it means, but I don\'t think that "a precise\n>concept" is what a mathematical statistician would call it.\n>\n>I am talking here (s.p.r.) physics language.\n>In mathematical terms, a classical ensemble is the set of elementary\n>events underlying the sigma algebra over which the measure is defined.\n\nI am afraid that this shows a SERIOUS misunderstanding of measure\ntheory (i.e. Borel sets and Lebesgue measure). Yes, discrete measures\n(i.e. over countable sets) have such a basis, but that does NOT extend\nto the general case. And using that \'simplification\' vastly complicates\nthe theory.\n\nIn general, there IS no set of elementary events underlying the Borel\nset. Even when that is defined on top of another set that does have\na concept of basic elements (which is not necessarily the case),\nit isn\'t rare for the measure of all such elements to be zero. The\nsimple and classic example is the real interval [0,1] with the uniform\nmeasure.\n\n\nRegards,\nNick Maclaren.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <4121F056.7090303@univie.ac.at>,
Arnold Neumaier <Arnold.Neumaier@univie.ac.atwrote:
>>>
>>>To say that "The probability that someone in risk group A will die
>>>of cancer is 1/3" means nothing more or less than that exactly 1/3
>>>of _all_ people in risk group A will die of cancer.
>>
>That is completely and utterly wrong. You can see that by taking
>a nice, simple example (i.e. not cancer).
>>
>Fair coins have a probability .5 of coming up heads. If you toss
>10 fair coins, it is NOT necessarily the case that exactly 5 will
>show heads.
>
>This is not a correct translation of my claim.

Good, Because it is completely wrong. It IS a correct example of
what you posted, so I am glad that you didn't mean it.

> >If you take any finite [itex]\sigma[/itex] algebra representing a
> >fair coin, one has a finite ensemble of elementary events,
> >and exactly half of them come out heads.
>
> If you take an infinite
[itex]>\sigma[/itex] algebra, the ensemble is infinite, but with the natural
>weighting, again exactly half of them come out head.

The very concept of "with the natural weighting, again exactly half
of them come out head" is misleading to a degree when applied to a
limit process (which this is). See below.

>'Tossing 10 fair coins' is just a sloppy way of saying
>'Selecting a sample of size 10 from the total ensemble',
>and it is obvious that here the number of heads is 5 only on
>average over many random samples, again as I had claimed in an
>unquoted part of the post to which you replied.
>
>>>On the other hands, "ensemble" is a precise concept with far-reaching
>>>applications, independent of any beliefs, and hence adequate for use
>>>in quantum mechanics.
>>
>Well, it wasn't a standard term when I did my (masters equivalent)
>course in mathematical statistics, though that was some 30+ years
>back. I can guess what it means, but I don't think that "a precise
>concept" is what a mathematical statistician would call it.
>
>I am talking here (s.p.r.) physics language.
>In mathematical terms, a classical ensemble is the set of elementary
>events underlying the [itex]\sigma[/itex] algebra over which the measure is defined.

I am afraid that this shows a SERIOUS misunderstanding of measure
theory (i.e. Borel sets and Lebesgue measure). Yes, discrete measures
(i.e. over countable sets) have such a basis, but that does NOT extend
to the general case. And using that 'simplification' vastly complicates
the theory.

In general, there IS no set of elementary events underlying the Borel
set. Even when that is defined on top of another set that does have
a concept of basic elements (which is not necessarily the case),
it isn't rare for the measure of all such elements to be zero. The
simple and classic example is the real interval [0,1] with the uniform
measure.

Regards,
Nick Maclaren.

Arnold Neumaier

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nDaryl McCullough wrote:\n> Arnold Neumaier says...\n>\n>\n>>Probability assignments to single events can be neither verified nor\n>>falsified.\n>\n> Probabilistic predictions can *never* be verified or falsified by any\n> (finite) number of observations. If the prediction is that half of all\n> particles of type X decay within T seconds, how many measurements does\n> it take to prove the prediction is true? How many measurements does\n> it take to prove the prediction is false?\n\nAll those in the defining ensemble. You seem to be thinking of an\ninfinite ensemble; then your statement is true. But if the ensemble\nis finite, one knows the probability of any statement about a random\nvariable x once all realizations x(omega) and their weight are known.\nThis completely characterizes the ensemble.\n\n\nArnold Neumaier\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Daryl McCullough wrote:
> Arnold Neumaier says...
>
>
>>Probability assignments to single events can be neither verified nor
>>falsified.
>
> Probabilistic predictions can *never* be verified or falsified by any
> (finite) number of observations. If the prediction is that half of all
> particles of type X decay within T seconds, how many measurements does
> it take to prove the prediction is true? How many measurements does
> it take to prove the prediction is false?

All those in the defining ensemble. You seem to be thinking of an
infinite ensemble; then your statement is true. But if the ensemble
is finite, one knows the probability of any statement about a random
variable x once all realizations [itex]x(\omega)[/itex] and their weight are known.
This completely characterizes the ensemble.

Arnold Neumaier

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