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Mar 22, 2004, 5:36:22 AM3/22/04

to

I was trying to figure out if the frequentist interpretation could be used

as the foundation of the probabilistic interpretation of QM. As I understand

(correct me if I'm wrong) the basis of the frequentist interpretation can be

summarized in the following statement:

We have an repeatable experiment with M possible outcomes, each with

probability P_m. Let's repeat the experiment N times. The number of times we

get the m'th result is X_m . The statement is that,

when N goes to infinity, X_m/N -> P_m.

You can take this as a definition of probability P_m.

But how is this limit defined? Does it mean that for every epsilon there is

a N(epsilon) such that for any n > N(epsilon) the value |X_m / n - P_m| <

epsilon ?

Can this N(epsilon) be calculated? Does it depend on the details of the

experiment?

This doesn't make sense to me, so I'm wondering if my intuition is

completely wrong.

Mar 22, 2004, 6:05:15 AM3/22/04

to

In experimental terms, this limit is undefinable since during the

sum of lifetimes of all physicists ever, only a finite number of

experiments have been performed. And I fear this will be the case

in the near future, too. Thus a limit makes sense only on the

theoretical level. But there is no problem with probabilities anyway.

You can find the justification of a relative frequency interpretation

in any textbook of probability under the heading of the weak law

of large numbers. The limit is 'in probability', which means that

the probability of violating |X_m / n - P_m| < epsilon goes to zero

as n gets large. How large n must be at a given confidence level

can be calculated, if one is careful in the argument leading to the

proof. Unfortunately there is nothing that excludes the unlikely

remaining probability...

Arnold Neumaier

Mar 23, 2004, 12:42:14 PM3/23/04

to

Arnold Neumaier <Arnold....@univie.ac.at> writes:

>Bartosz Milewski wrote:

>> I was trying to figure out if the frequentist interpretation could be used

>> as the foundation of the probabilistic interpretation of QM. ...

> ...

>You can find the justification of a relative frequency interpretation

>in any textbook of probability under the heading of the weak law

>of large numbers. The limit is 'in probability', which means that

>the probability of violating |X_m / n - P_m| < epsilon goes to zero

>as n gets large. How large n must be at a given confidence level

>can be calculated, if one is careful in the argument leading to the

>proof. Unfortunately there is nothing that excludes the unlikely

>remaining probability...

Right, so actually, the frequentist interpretation of probability

suffers from the same disease that the many-worlds interpretation

does, or at least the non-Bayesian one. In many worlds, the problem

is that there's no way to justify dismissing worlds with a small

quantum amplitude as being rare, and in the frequentist

version of probability theory, there's no way to justify dismissing

outcomes with small probability as being rare.

The frequentist interpretation of probability suffers from worse

diseases as well. For example, you'll find in many probability

books and hear from the mouths of top probability theorists the

claim that no process can produce random, uniformly distributed

positive integers, but that processes can produce random uniformly

distributed real numbers between zero and one (e.g. toss a fair

coin exactly aleph_0 times to get the binary expansion).

In fact, a process which produces uniformly distributed random real

numbers between zero and one can be modified so that it produces

uniformly distributed random positive integers in the following

way: Consider [0,1) as an additive group of reals modulo 1. Then

it has a subgroup, S, consisting of rational numbers in [0,1). Form

a set X by choosing one element from each coset of S in [0,1). Then

define X_r = {a+r mod 1 | a \in X}, for each r in S. The X_r are

pairwise disjoint, pairwise congruent sets, with congruent meaning

they are related to each other by isometries of the group [0,1).

In that sense, they are as equiprobable as can be. Now if q is a

random number between 0 and 1, then it falls into exactly one X_r,

so there is a unique rational number, r, associated with that real

number, and since the rationals are countable, there is also a

unique positive integer associated with that real number. Since the

X_r's are congruent, no one can be any more or less likely than any

other, so no positive integer is any more or less likely than any

other to result from this process. Voila, we have a way to get a

"random" positive integer from a "random" real in [0,1).

The problem is that if you define probabilities in terms of

outcomes of repeated processes or experiments, then you might get

lead astray when you find that certain probability distributions

don't exist (e.g. a uniform distribution over the positive

integers). You might start imagining that your probability theory

is telling you something about what kind of random-number-generation

processes are or aren't possible. As the example above shows, this

is incorrect.

R.

Ps. Yes, I know I used the axiom of choice and rely on axioms of

infinity, but that's not a problem. Nobody would actually

drop these axioms in order to save the frequentist interpretation,

or at least, nobody worth mentioning.

Mar 24, 2004, 9:45:31 PM3/24/04

to

"Arnold Neumaier" <Arnold....@univie.ac.at> wrote in message

news:405EC787...@univie.ac.at...

> You can find the justification of a relative frequency interpretation

> in any textbook of probability under the heading of the weak law

> of large numbers. The limit is 'in probability', which means that

> the probability of violating |X_m / n - P_m| < epsilon goes to zero

> as n gets large. How large n must be at a given confidence level

> can be calculated, if one is careful in the argument leading to the

> proof. Unfortunately there is nothing that excludes the unlikely

> remaining probability...

news:405EC787...@univie.ac.at...

> You can find the justification of a relative frequency interpretation

> in any textbook of probability under the heading of the weak law

> of large numbers. The limit is 'in probability', which means that

> the probability of violating |X_m / n - P_m| < epsilon goes to zero

> as n gets large. How large n must be at a given confidence level

> can be calculated, if one is careful in the argument leading to the

> proof. Unfortunately there is nothing that excludes the unlikely

> remaining probability...

Doesn't this justification suffer from the same problem? What's the precise

meaning of "probability goes to zero as n gets large"? I don't know if

statements like this have a meaning at all. In mathematics "goes to" or "has

a limit" are very well defined notions (i.e., for any epsilon there exist...

etc.).

I can think of an unorthodox way of dealing with limits in probability.

Introduce the "random number generator," just like in computer simulations

(in practice one uses a pseudo-random generator). The N(epsilon) such that

for each n > N(epsilon) |X_m / n - P_m| < epsilon could then be calculated

based on the properties of the "random" number stream that generates values

X_m.

Here's some handwaving: Imagine that a stream of "random" numbers has the

following property: If you generate heads and tails using this stream, then

there is a certain N_1 above which you are guaranteed that the first N

tosses could not have been all heads or all tails (there is at least 1 odd

result). I don't know how much this would break radnomness properties, but

it would make the definition of limits meaningful.

By the way, has anyone tested experimetally the randomness of quantum

experiments? Is a quantum random number generator perfectly random?

Mar 25, 2004, 5:04:00 AM3/25/04

to

Bartosz Milewski <bar...@nospam.relisoft.com> wrote:

> By the way, has anyone tested experimetally the randomness of quantum

> experiments? Is a quantum random number generator perfectly random?

Of course not, perfection doesn't exist.

There are always instrumental effects that spoil it.

A well known, and already very old example

is the dead time of Geiger counters.

This spoils the ideal Poisson distribution of the famous clicks.

For a practical implementation of a hardware random number generator

one often uses Zener diodes biassed in the reverse direction.

Since the current is caused by tunnelling of electrons through a barrier

this may wel be considered to be a quantummechanical noise source.

Suitable electronics can transform the variable current

into (very nearly) random bits, hence numbers.

Some small bias remains though.

Jan

Mar 25, 2004, 8:25:50 PM3/25/04

to

"Bartosz Milewski" <bar...@nospam.relisoft.com> wrote in message news:<c3qnae$scv$1...@brokaw.wa.com>...

>

>

> Here's some handwaving: Imagine that a stream of "random" numbers has the

> following property: If you generate heads and tails using this stream, then

> there is a certain N_1 above which you are guaranteed that the first N

> tosses could not have been all heads or all tails (there is at least 1 odd

> result).

But this is exactly what the original question was all about:

the frequentist application of probability theory doesn't make sense

if there is no "lower cutoff" below which we consider that the event

will never happen and which is NOT 0.

Indeed, naively said, the law of LARGE numbers cannot be replaced by

the law of INFINITE numbers because then probability theory makes NO

statement at all about finite statistics.

If you toss a coin N times, the probability to have all heads is

1/2^N. So if you toss a coin N times, to estimate the probability of

having "heads" on one go (which you expect to be 1/2), there is a

probability of 1/2^N that you will actually find 1 (and also a

probability of 1/2^N that you will find 0).

If N = 10000, then that's a small probability indeed, but according to

orthodox probability theory, it CAN occur.

Now let us say that we repeat this tossing of N coins M times, you'd

then expect that finding an estimate of the probability of one tossing

of the coin = 1 will only occur, on average, 1/2^N times M. But there

is again, a very small probability that we will find for ALL of these

M experiments (each with N coins), an average of 1 (namely 1/2^(NxM)

is that probability).

So the statement that "events with an extremely small probability

associated to it will probably not occur" is an empty statement

because tautological. It is only when we say that "events with an

extremely small probability will NOT occur" that suddenly, all of the

frequentist interpretation of probability theory makes sense. But

that statement is a strong one, however, in practical, and

experimental life we always make it. Most, if not all, experimental

claims are associated with a small probability (10 sigma for example)

that it is a statistical fluctuation, but beyond a certain threshold,

people take it as a hard fact.

cheers,

Patrick.

Mar 25, 2004, 8:26:24 PM3/25/04

to

Bartosz Milewski <bar...@nospam.relisoft.com> wrote:

Your conceptual problem has nothing to do with quantum mechanics.

It arises in precisely the same form when you want to verify by

experiment that a coin being thrown repeatedly is fair

(That is, has exactly 50% probability of coming up heads or tails)

For the resolution see any textbook on probability:

you can never verify such a thing, you can only give confidence limits.

It has nothing to do with frequentist versus Bayesianism either:

a Bayesian can do no better.

Best,

Jan

Mar 26, 2004, 2:37:42 AM3/26/04

to

> You can find the justification of a relative frequency interpretation

> in any textbook of probability under the heading of the weak law

> of large numbers. The limit is 'in probability', which means that

> the probability of violating |X_m / n - P_m| < epsilon goes to zero

> as n gets large. How large n must be at a given confidence level

> can be calculated, if one is careful in the argument leading to the

> proof. Unfortunately there is nothing that excludes the unlikely

> remaining probability...

> in any textbook of probability under the heading of the weak law

> of large numbers. The limit is 'in probability', which means that

> the probability of violating |X_m / n - P_m| < epsilon goes to zero

> as n gets large. How large n must be at a given confidence level

> can be calculated, if one is careful in the argument leading to the

> proof. Unfortunately there is nothing that excludes the unlikely

> remaining probability...

Notice that the argument is circular, i.e. one uses a concept of

probability in order to define the concept of probability. This

doesn't cause problems for most applications of probability theory,

but it is the main reason to be a Bayesian from the conceptual point

of view.

Mar 26, 2004, 4:49:59 AM3/26/04

to

Bartosz Milewski wrote

> has anyone tested experimentally the randomness of quantum

> experiments?

Try the e-print physics/0304013. Here's the abstract:

*******

physics/0304013

From: Dana J. Berkeland [view email]

Date: Fri, 4 Apr 2003 18:16:19 GMT (143kb)

Tests for non-randomness in quantum jumps

Authors: D.J. Berkeland, D.A. Raymondson, V.M. Tassin

Comments: 4 pages, 5 figures

Subj-class: Atomic Physics

In a fundamental test of quantum mechanics, we have collected

over 250,000 quantum jumps from single trapped and cooled

88Sr+ ions, and have tested their statistics using a

comprehensive set of measures designed to detect non-random

behavior. Furthermore, we analyze 238,000 quantum jumps from

two simultaneously confined ions and find that the number of

apparently coincidental transitions is as expected. Similarly, we

observe 8400 spontaneous decays of two simultaneously trapped

ions and find that the number of apparently coincidental decays

agrees with expected value. We find no evidence for short- or

long-term correlations in the intervals of the quantum jumps or

in the decay of the quantum states, in agreement with quantum

theory.

*****

Matthew Donald (matthew...@phy.cam.ac.uk)

web site:

http://www.poco.phy.cam.ac.uk/~mjd1014

``a many-minds interpretation of quantum theory''

*****************************************

Mar 26, 2004, 4:50:02 AM3/26/04

to

Bartosz Milewski wrote:

> "Arnold Neumaier" <Arnold....@univie.ac.at> wrote in message

> news:405EC787...@univie.ac.at...

>

>>You can find the justification of a relative frequency interpretation

>>in any textbook of probability under the heading of the weak law

>>of large numbers. The limit is 'in probability', which means that

>>the probability of violating |X_m / n - P_m| < epsilon goes to zero

>>as n gets large. How large n must be at a given confidence level

>>can be calculated, if one is careful in the argument leading to the

>>proof. Unfortunately there is nothing that excludes the unlikely

>>remaining probability...

>

> Doesn't this justification suffer from the same problem? What's the precise

> meaning of "probability goes to zero as n gets large"?

It has a precise mathematical meaning. it has no precise physical meaning.

But nothing at all has a precise physical meaning. Only theory can be

exact - you can say exactly what an electron is in QED; you cannot

say exactly what it is in reality.

The interface between theory and the real world is never exact.

This interface must just be clear enough to guarantee working protocols

for the execution of science in practice.

I think it is not reasonable to require of probability a more stringent

meaning than for an electron. It suffices that there are recipes that

give, in practice acceptable approximations.

Arnold Neumaier

Mar 26, 2004, 5:49:48 AM3/26/04

to

Matt Leifer wrote:

>>You can find the justification of a relative frequency interpretation

>>in any textbook of probability under the heading of the weak law

>>of large numbers. The limit is 'in probability', which means that

>>the probability of violating |X_m / n - P_m| < epsilon goes to zero

>>as n gets large. How large n must be at a given confidence level

>>can be calculated, if one is careful in the argument leading to the

>>proof. Unfortunately there is nothing that excludes the unlikely

>>remaining probability...

>

>

> Notice that the argument is circular, i.e. one uses a concept of

> probability in order to define the concept of probability.

No. It uses the concept of probability to explain why, within this

framework, relative frequences are valid approximations to

probabilities. Probabilities themselves are defined axiomatically,

and not justified at all.

Any theory needs to start somewhere with some basic, unexplained

terms that get their meaning from the consequences of the theory,

and not from anything outside.

Arnold Neumaier

Mar 26, 2004, 5:49:51 AM3/26/04

to

Patrick Van Esch wrote:

> So the statement that "events with an extremely small probability

> associated to it will probably not occur" is an empty statement

> because tautological.

Yes.

> It is only when we say that "events with an

> extremely small probability will NOT occur"

but this is wrong. If you randomly draw a real number x from a unifrom

distribution in [0,1], and get as a result s, the probability that

you obtained exactly this number is zero, but it was the one you got.

> that suddenly, all of the

> frequentist interpretation of probability theory makes sense.

This has nothing to do with the frequentist interpretation; the

problem of unlikely things happening is also present in a Bayesian

interpretation.

The problem stems from the attempt to assign 100% precise meaning

to concepts that have an inherent uncertainty. Once one realizes

that any concept applied to reality is limited since we cannot

say too precisely what is meant by it on the operational level

(100% clarity is avaialble only in theoretical models), the

difficulty disappears. That's why it has become standard practice

to distinguish between reality and our models of it.

Arnold Neumaier

Mar 26, 2004, 5:24:30 PM3/26/04

to

"Patrick Van Esch" <van...@ill.fr> wrote in message

news:c23e597b.04032...@posting.google.com...

> So the statement that "events with an extremely small probability

> associated to it will probably not occur" is an empty statement

> because tautological. It is only when we say that "events with an

> extremely small probability will NOT occur" that suddenly, all of the

> frequentist interpretation of probability theory makes sense.

news:c23e597b.04032...@posting.google.com...

> So the statement that "events with an extremely small probability

> associated to it will probably not occur" is an empty statement

> because tautological. It is only when we say that "events with an

> extremely small probability will NOT occur" that suddenly, all of the

> frequentist interpretation of probability theory makes sense.

Yes, that's exactly my point. I was trying to make the cutoff somewhat

better defined making it a property of a "random" number generator. Let me

clarify my point: In mathematics, probability theory describes properties of

"measures." It's a self-consistent theory (modulo Goedel) and that's it. In

physics we are trying to interpret these measures as probabilities, so we

have to provide a framework. The frequentist framework doesn't seem to be

consistent. I propose to extend the frequentist interpretation by

abstracting the random number generator part of it. Probability can then be

defined formally as a limit of frequencies, provided the frequencies fulfill

some additional properties--the cutoff properties. In essence, they must

behave as if they were generated by a computer program using a random number

generator with some well defined cutoff property. This random number

generator is necessary so that all the frequencies exhibit the same cutoff

property (i.e., the frequency of [quantum] heads has the same cutoffs as the

frequency of tails).

Mar 26, 2004, 5:21:56 PM3/26/04

to

"J. J. Lodder" <nos...@de-ster.demon.nl> wrote in message

news:1gb70ve.44...@de-ster.xs4all.nl...

> Bartosz Milewski <bar...@nospam.relisoft.com> wrote:

>

> Your conceptual problem has nothing to do with quantum mechanics.

> It arises in precisely the same form when you want to verify by

> experiment that a coin being thrown repeatedly is fair

> (That is, has exactly 50% probability of coming up heads or tails)

news:1gb70ve.44...@de-ster.xs4all.nl...

> Bartosz Milewski <bar...@nospam.relisoft.com> wrote:

>

> Your conceptual problem has nothing to do with quantum mechanics.

> It arises in precisely the same form when you want to verify by

> experiment that a coin being thrown repeatedly is fair

> (That is, has exactly 50% probability of coming up heads or tails)

There is a huge difference between quantum probability and classical

probability. Coin tosses are not "really" random. They are chaotic, which

means we can't predict the results because (a) we never know the initial

conditions _exactly_ (the butterfly effect) and (b) because we don't have

computers powerful enough to model a coin toss. So coin tossing is for all

"practical" purposes random, but theoreticall it's not! In QM, on the other

hand, randomness is inherent. If you can prepare a system in pure state, you

know the initial conditions _exactly_. And yet, the results of experiments

are only predicted probabilistically. Moreover, there are no hidden

variables (this approach has been tried), whose knowledge could specify the

initial conditions more accurately and maybe let you predict the exact

outcomes.

I have no problems with coin tosses as long as you don't use a quantum coin.

Mar 27, 2004, 6:15:11 AM3/27/04

to

"Arnold Neumaier" <Arnold....@univie.ac.at> wrote in message

news:40640905...@univie.ac.at...

> but this is wrong. If you randomly draw a real number x from a unifrom

> distribution in [0,1], and get as a result s, the probability that

> you obtained exactly this number is zero, but it was the one you got.

This argument is also circular. How do you draw a number randomly? It's not

a facetious question.

Mar 29, 2004, 2:36:38 AM3/29/04

to

Arnold Neumaier <Arnold....@univie.ac.at> wrote in message news:<40640905...@univie.ac.at>...

> Patrick Van Esch wrote:

>

> > So the statement that "events with an extremely small probability

> > associated to it will probably not occur" is an empty statement

> > because tautological.

>

> Yes.

>

> > It is only when we say that "events with an

> > extremely small probability will NOT occur"

>

> but this is wrong. If you randomly draw a real number x from a unifrom

> distribution in [0,1], and get as a result s, the probability that

> you obtained exactly this number is zero, but it was the one you got.

> Patrick Van Esch wrote:

>

> > So the statement that "events with an extremely small probability

> > associated to it will probably not occur" is an empty statement

> > because tautological.

>

> Yes.

>

> > It is only when we say that "events with an

> > extremely small probability will NOT occur"

>

> but this is wrong. If you randomly draw a real number x from a unifrom

> distribution in [0,1], and get as a result s, the probability that

> you obtained exactly this number is zero, but it was the one you got.

I'm not talking about the mathematical theory of probabilities (a la

Kolmogorov) which is a nice mathematical theory. I'm talking about

the application of this mathematical theory to the physical sciences,

which maps events (potential experimental outcomes) to numbers (called

probabilities, but we could call it "gnorck"). Saying that measuring

the number of neutrons scattered under an angle of 12 to 13 degrees

has gnorck = 10^(-6) doesn't mean much as such. So this assignment of

probabilities to experimental outcomes has only a meaning when it

eventually turns into "hard" statements, and that can only happen when

we apply a lower cutoff to probabilities.

Your argument of drawing a real number out of [0,1] doesn't apply

here, because the outcome of an experiment is never a true real number

(most of which cannot even be written down !). There are always a

finite number of possibilities in the outcome of an experiment

(otherwise it couldn't be written onto a hard disk!).

cheers,

Patrick.

Mar 29, 2004, 4:22:03 AM3/29/04

to

"Bartosz Milewski" <bar...@nospam.relisoft.com> wrote in message news:<c426a4$bvd$1...@brokaw.wa.com>...

> "J. J. Lodder" <nos...@de-ster.demon.nl> wrote in message

> news:1gb70ve.44...@de-ster.xs4all.nl...

> > Bartosz Milewski <bar...@nospam.relisoft.com> wrote:

> >

> > Your conceptual problem has nothing to do with quantum mechanics.

> > It arises in precisely the same form when you want to verify by

> > experiment that a coin being thrown repeatedly is fair

> > (That is, has exactly 50% probability of coming up heads or tails)

>

> There is a huge difference between quantum probability and classical

> probability. Coin tosses are not "really" random. They are chaotic, which

> means we can't predict the results because (a) we never know the initial

> conditions _exactly_ (the butterfly effect) and (b) because we don't have

> computers powerful enough to model a coin toss.

That "we can't predict the results" IS randomness. There is nothing

more to randomness than that.

Regards,

IV

Mar 29, 2004, 5:55:17 AM3/29/04

to

Bartosz Milewski wrote:

> There is a huge difference between quantum probability and classical

> probability. Coin tosses are not "really" random.

What is "really" random??? the term "random" has no precise meaning

outside theory. But in the theory of stochastic processes, there is

"real" classical randomness.

> In QM, on the other

> hand, randomness is inherent. If you can prepare a system in pure state, you

But no one can. Pure states in quantum mechanics are as much idealizations

as classical random processes.

Arnold Neumaier

Mar 29, 2004, 2:38:36 PM3/29/04

to

Patrick Van Esch wrote:

> Arnold Neumaier <Arnold....@univie.ac.at> wrote in message news:<40640905...@univie.ac.at>...

>

>>Patrick Van Esch wrote:

> Arnold Neumaier <Arnold....@univie.ac.at> wrote in message news:<40640905...@univie.ac.at>...

>

>>Patrick Van Esch wrote:

>>>It is only when we say that "events with an

>>>extremely small probability will NOT occur"

>>

>>but this is wrong. If you randomly draw a real number x from a unifrom

>>distribution in [0,1], and get as a result s, the probability that

>>you obtained exactly this number is zero, but it was the one you got.

>

> I'm not talking about the mathematical theory of probabilities (a la

> Kolmogorov) which is a nice mathematical theory. I'm talking about

> the application of this mathematical theory to the physical sciences,

> which maps events (potential experimental outcomes) to numbers (called

> probabilities, but we could call it "gnorck").

In that case, all statements are approximate statements only,

and probability has no exact meaning. Saying p=1e-12

in a practical situation where you can never collect more than

a few thousands cases is as meaningless as saying that the

distance earth-moon is 384001.1283564032984930201549807km

> Saying that measuring

> the number of neutrons scattered under an angle of 12 to 13 degrees

> has gnorck = 10^(-6) doesn't mean much as such. So this assignment of

> probabilities to experimental outcomes has only a meaning when it

> eventually turns into "hard" statements, and that can only happen when

> we apply a lower cutoff to probabilities.

But where exactly would you take it? If there is an eps such that

p<eps means 'it will not occur', what is the supremum of all such eps?

It would have to be a fundamental constant of nature.

The nonexistence of such a constant implies that your proposal cannot

be right.

> Your argument of drawing a real number out of [0,1] doesn't apply

> here, because the outcome of an experiment is never a true real number

> (most of which cannot even be written down !). There are always a

> finite number of possibilities in the outcome of an experiment

> (otherwise it couldn't be written onto a hard disk!).

This does not really help. Let eps be the "extremely small probability"

according to your proposal. Pick N >> - log eps / log 2, and

run a series of N coin tosses. You get the result x_1 ... x_N, say.

Although you really obtained exactly this result, the probability of

obtaining it was only 2^{-N} << eps. Thus your proposal amounts to

proving the impossibility of tossing coins more than a fairly small

number of times... Do you really want to claim that???

The truth is that probabilities, just like real numbers,

are concepts of theory, and as such apply only approximately

to reality, with a context-dependent and user-dependent accuracy

with fuzzy boundaries.

Arnold Neumaier

Mar 30, 2004, 3:47:24 AM3/30/04

to

r...@maths.tcd.ie wrote in message news:<c3njd0$295l$1...@lanczos.maths.tcd.ie>...

> Arnold Neumaier <Arnold....@univie.ac.at> writes:

>

>

> >Bartosz Milewski wrote:

> >> I was trying to figure out if the frequentist interpretation could be used

> >> as the foundation of the probabilistic interpretation of QM. ...

>

> > ...

>

> >You can find the justification of a relative frequency interpretation

> >in any textbook of probability under the heading of the weak law

> >of large numbers. The limit is 'in probability', which means that

> >the probability of violating |X_m / n - P_m| < epsilon goes to zero

> >as n gets large. How large n must be at a given confidence level

> >can be calculated, if one is careful in the argument leading to the

> >proof. Unfortunately there is nothing that excludes the unlikely

> >remaining probability...

The idea is that the probability may be made as small as one likes.

So it can be made so small that the event is for all practical

purposes impossible.

>

> Right, so actually, the frequentist interpretation of probability

> suffers from the same disease that the many-worlds interpretation

> does, or at least the non-Bayesian one. In many worlds, the problem

> is that there's no way to justify dismissing worlds with a small

> quantum amplitude as being rare, and in the frequentist

> version of probability theory, there's no way to justify dismissing

> outcomes with small probability as being rare.

>

Quantum theory is a probabilistic theory and extremely unlikely events

are not excluded, nor should they be. So this is a property of the

theory, not the interpretation. It seems to me that an interpretation

that excluded such events absolutely would be in error.

> Arnold Neumaier <Arnold....@univie.ac.at> writes:

>

>

> >Bartosz Milewski wrote:

> >> I was trying to figure out if the frequentist interpretation could be used

> >> as the foundation of the probabilistic interpretation of QM. ...

>

> > ...

>

> >You can find the justification of a relative frequency interpretation

> >in any textbook of probability under the heading of the weak law

> >of large numbers. The limit is 'in probability', which means that

> >the probability of violating |X_m / n - P_m| < epsilon goes to zero

> >as n gets large. How large n must be at a given confidence level

> >can be calculated, if one is careful in the argument leading to the

> >proof. Unfortunately there is nothing that excludes the unlikely

> >remaining probability...

So it can be made so small that the event is for all practical

purposes impossible.

>

> Right, so actually, the frequentist interpretation of probability

> suffers from the same disease that the many-worlds interpretation

> does, or at least the non-Bayesian one. In many worlds, the problem

> is that there's no way to justify dismissing worlds with a small

> quantum amplitude as being rare, and in the frequentist

> version of probability theory, there's no way to justify dismissing

> outcomes with small probability as being rare.

>

are not excluded, nor should they be. So this is a property of the

theory, not the interpretation. It seems to me that an interpretation

that excluded such events absolutely would be in error.

> The frequentist interpretation of probability suffers from worse

> diseases as well. For example, you'll find in many probability

> books and hear from the mouths of top probability theorists the

> claim that no process can produce random, uniformly distributed

> positive integers, but that processes can produce random uniformly

> distributed real numbers between zero and one (e.g. toss a fair

> coin exactly aleph_0 times to get the binary expansion).

Yes these claims as stated are contradictory. I suspect that the

definitions you are using are imprecise. The word "process" implies

computability, that the process is finite. A real number is cleverly

defined as a limit of a finite process. So a real number is

computable in this sense, that it can be approximated as closely as

one likes in finite time. The problem with your proof is that as the

real number is computed the choice of cosets changes with each step so

the process does not converge to an integer.

Using the axioms of choice and infinity then one can indeed choose a

natural number at random. There are some rather strange consequences.

It is then possible to prove that each number chosen in this way will

be greater than all such previously chosen numbers with probability

one. Let N be the greatest such number chosen so far. Then there are

finitely many natural numbers less than or equal to N but infinitely

many greater than N. So the next number chosen will be greater than N

with probability one. Note that our ostensibly random sequence is

strictly increasing with probability one. This is not the only

bizarre consequence of the axiom of choice: see the well-known

Banach-Tarski sphere paradox. So I should think a physicist would do

well to be wary of the axiom of choice as tending to produce

non-physical results.

The frequentist approach does not assume the axiom of choice and makes

no use of transfinite mathematics or completed limits. If it did, the

problems you mention would in fact arise.

Mar 30, 2004, 11:32:30 AM3/30/04

to

>>For example, you'll find in many probability

>>books and hear from the mouths of top probability theorists the

>>claim that no process can produce random, uniformly distributed

>>positive integers, but that processes can produce random uniformly

>>distributed real numbers between zero and one (e.g. toss a fair

>>coin exactly aleph_0 times to get the binary expansion).

This has a very simple reason: There is no consistent definition of

random, uniformly distributed positive integers, while there is

one for random uniformly distributed real numbers between zero and one.

This is a purely mathematical statement independent of any

interpretation!

And of course, when people say 'produce' they mean

'produce in theory', or if they mean 'produce in practice' they

have in mind that it is produced only approximately.

Arnold Neumaier

Mar 30, 2004, 12:42:53 PM3/30/04

to

Patrick Powers wrote:

>

> r...@maths.tcd.ie wrote in message news:<c3njd0$295l$1...@lanczos.maths.tcd.ie>...

> > Arnold Neumaier <Arnold....@univie.ac.at> writes:

[snip]

> Using the axioms of choice and infinity then one can indeed choose a

> natural number at random. There are some rather strange consequences.

> It is then possible to prove that each number chosen in this way will

> be greater than all such previously chosen numbers with probability

> one. Let N be the greatest such number chosen so far. Then there are

> finitely many natural numbers less than or equal to N but infinitely

> many greater than N. So the next number chosen will be greater than N

> with probability one. Note that our ostensibly random sequence is

> strictly increasing with probability one.

Hmm, interesting post, thanks.

This is not the only

> bizarre consequence of the axiom of choice: see the well-known

> Banach-Tarski sphere paradox. So I should think a physicist would do

> well to be wary of the axiom of choice as tending to produce

> non-physical results.

But this only happens if you use it in a scenario that is *already*

unphysical, e.g. if you claim it's possible to draw a lottery ball

from a cage containing aleph_0 ping-pong balls. Here the physical

problem is not in the drawing, but in the setting up of the cage to

begin with. As for Banach-Tarski, it also requires an unphysical

scenario; you can't make it work by pulverizing and reassembling a

ball made of any physical material.

So I don't see any problem with a physicist accepting AC and its

useful consequences as applied to the mathematics of continua.

Mar 30, 2004, 12:29:07 PM3/30/04

to

Bartosz Milewski <bar...@nospam.relisoft.com> wrote:

> "J. J. Lodder" <nos...@de-ster.demon.nl> wrote in message

> news:1gb70ve.44...@de-ster.xs4all.nl...

> > Bartosz Milewski <bar...@nospam.relisoft.com> wrote:

> >

> > Your conceptual problem has nothing to do with quantum mechanics.

> > It arises in precisely the same form when you want to verify by

> > experiment that a coin being thrown repeatedly is fair

> > (That is, has exactly 50% probability of coming up heads or tails)

>

> There is a huge difference between quantum probability and classical

> probability.

Not at all, from the point of view of probability theory.

> Coin tosses are not "really" random. They are chaotic, which

> means we can't predict the results because (a) we never know the initial

> conditions _exactly_ (the butterfly effect) and (b) because we don't have

> computers powerful enough to model a coin toss. So coin tossing is for all

> "practical" purposes random, but theoreticall it's not!

We use probability theory when we don't know, or don't want to know,

about underlying causes.

Whether or not such causes are actually present is irrelevant.

Probability just deals with 'something' that produces heads or tails,

and determines properties of the sequence of them,

(like confidence in being fair)

using the means of probability theory.

> In QM, on the other

> hand, randomness is inherent. If you can prepare a system in pure state, you

> know the initial conditions _exactly_. And yet, the results of experiments

> are only predicted probabilistically. Moreover, there are no hidden

> variables (this approach has been tried), whose knowledge could specify the

> initial conditions more accurately and maybe let you predict the exact

> outcomes.

>

> I have no problems with coin tosses as long as you don't use a quantum coin.

All coins are quantum coins, for we live in a quantum word.

In practice it may be quite hard to say whether or not

a coin throw may be considered to be 'classical'.

Quantum mechanics may come in in the precise timing

of the twitching of your fingers, on the molecular level,

when flipping the coin.

Not that it matters,

Jan

Mar 30, 2004, 12:31:45 PM3/30/04

to

"Bartosz Milewski" <bar...@nospam.relisoft.com> wrote in message news:<c428jd$cvv$1...@brokaw.wa.com>...

> "Patrick Van Esch" <van...@ill.fr> wrote in message

> news:c23e597b.04032...@posting.google.com...

> > So the statement that "events with an extremely small probability

> > associated to it will probably not occur" is an empty statement

> > because tautological. It is only when we say that "events with an

> > extremely small probability will NOT occur" that suddenly, all of the

> > frequentist interpretation of probability theory makes sense.

>

> Yes, that's exactly my point. I was trying to make the cutoff somewhat

> better defined making it a property of a "random" number generator.

> "Patrick Van Esch" <van...@ill.fr> wrote in message

> news:c23e597b.04032...@posting.google.com...

> > So the statement that "events with an extremely small probability

> > associated to it will probably not occur" is an empty statement

> > because tautological. It is only when we say that "events with an

> > extremely small probability will NOT occur" that suddenly, all of the

> > frequentist interpretation of probability theory makes sense.

>

> Yes, that's exactly my point. I was trying to make the cutoff somewhat

> better defined making it a property of a "random" number generator.

In fact, this point is something that bothered me since I learned

about probability theory (20 years ago) and most people didn't seem to

even understand what my problem was (depends probably on the people

you talk to).

The fact that others here seem to struggle with the same problem

indicates that this is somehow a problem :-)

However, I have no idea if you can make a consistent probability

interpretation with such a cutoff. I don't know if ever some work in

that direction has been undertaken.

The Bayesian interpretation of probabilities is a nice information

theoretical construct of course, I'm not expert enough in it to see if

the problem also exists there. But I have difficulties with people

who deny the frequentist interpretation: after all, this is - to me -

the only way to make a connection to experimental results ! How do

you verify differential cross sections ? You do a number of

experiments, and then you make the HISTOGRAM (counting the number of

occurences) of the outcomes which you compare with your calculated

probability density. That's nothing else but applying the frequentist

interpretation, no ?

cheers,

Patrick.

Mar 30, 2004, 12:44:24 PM3/30/04

to

I am not circular since I didn't attempt to give a definition of

probability but simply assumed it to refute a claim made.

If randomness has any meaning at all in practice, it must be able to

draw numbers randlomly; if not by the uniform distribution then by

whatever process is assumed.

If you accept that there is something like a 'fair coin toss'

which gives independent events with probability 1/2, you can

easily get arbitrary small probabilities without invoking real numbers,

as I showed in another reply in this thread.

If you can't accept a fair coin toss, I wonder whether you have

any place at all for probabilisitic models in your world view.

The problem in all these issues related to probability is the silent

switch between theory and practice at some place, which different

people take at a different place, which makes communication difficult

and invites paradoxes. The interface between theory and reality is

always a little vague, and one has to be careful not to make statements

which are meaningless.

Raw reality has no concepts; it simply is. But to do science,

and indeed already to live intelligently, one needs to sort reality

into various conceptual bags that allow one to understand and predict.

Because of our incomplete access to reality, we can do this only in an

imperfect, somewhat fuzzy way. Respecting this in one's thinking

avoids all paradoxes; coming across a paradox means that one moved

somewhere across the border of what was permitted - though it is not

always easy to see where and why.

Full clarity can be obtained only on the logical level, this is

necessarily one level away from reality. In the foundations of

logic, one builds within an intuitive logic a complete model of

everything logic is about, and then is able to clarify the limits

of logical reasoning. This is the closest one can get to a clear

understanding of the foundations. To do the same in physics amounts

to building within mathematics a complete model with all the features

external reality is believed to have, and to discuss within this model

all the concepts and activities physicists work with. If this can be

done in a consistent way, it is as close as we can get to ascertaining

that the model is indeed faithful to external reality.

Therefore, when _I_ discuss probability, I choose such a model

as the background on the basis of which I can speak of well-defined

probabilities. In such a mathematical world, one can draw number

by decree (even though one cannot know what was drawn).

In reality, one needs to substitute pseudo-random number generators,

which makes drawing random numbers a practical activity. But of

course their properties only approximate the theoretical thing,

as always when a formal concept is implemented in nature.

Not even the Peano axioms for natural numbers can be realized in

nature - how much less more subtle concepts like probability.

Arnold Neumaier

Mar 30, 2004, 12:48:40 PM3/30/04

to

news:4067FA65...@univie.ac.at...

> What is "really" random??? the term "random" has no precise meaning

> outside theory. But in the theory of stochastic processes, there is

> "real" classical randomness.

> What is "really" random??? the term "random" has no precise meaning

> outside theory. But in the theory of stochastic processes, there is

> "real" classical randomness.

But stochastic theory is NOT a fundamental theory. It's an idealization of a

chaotic process. Can we interpret QM as an idealization of some more

fundamental theory? A theory where events are theoretically 100%

predictable, but so chaotic that in practice we can only make stochastic

predictions? I'm afraid this path had been tried (hidden variables) and

rebuked.

Mar 30, 2004, 2:28:14 PM3/30/04

to

vec...@weirdtech.com (Italo Vecchi) wrote in message news:<61789046.04032...@posting.google.com>...

>

> That "we can't predict the results" IS randomness. There is nothing

> more to randomness than that.

>

>

> That "we can't predict the results" IS randomness. There is nothing

> more to randomness than that.

>

True. But some posters are saying that quantum processes are

essentially random in that asuming that the results can be predicted

leads to a contradiction. In other words, they say that it has been

proved impossible that the events will ever be predicted.

I know about Bell's theorem, but don't know that such a thing has been

proved in general. Can someone please provide a reference?

Mar 30, 2004, 2:28:20 PM3/30/04

to

Arnold Neumaier <Arnold....@univie.ac.at> wrote in message news:<406803CA...@univie.ac.at>...

> Patrick Van Esch wrote:

> > Arnold Neumaier <Arnold....@univie.ac.at> wrote in message news:<40640905...@univie.ac.at>...

> >

>

> Patrick Van Esch wrote:

> > Arnold Neumaier <Arnold....@univie.ac.at> wrote in message news:<40640905...@univie.ac.at>...

> >

>

> But where exactly would you take it? If there is an eps such that

> p<eps means 'it will not occur', what is the supremum of all such eps?

> It would have to be a fundamental constant of nature.

>

> The nonexistence of such a constant implies that your proposal cannot

> be right.

>

> p<eps means 'it will not occur', what is the supremum of all such eps?

> It would have to be a fundamental constant of nature.

>

> The nonexistence of such a constant implies that your proposal cannot

> be right.

>

I agree that it should be a constant of this universe, if ever there

was such a thing. It is not unthinkable that there IS such a constant

(such as the inverse of the number of spacetime events, which could be

finite). But I do realise how speculative that statement is.

Nevertheless, without this vague idea, I cannot make any sense of what

it experimentally means to have an event with a probability p. Or

better, why every time we do statistics on real data, it works out !

>

> > Your argument of drawing a real number out of [0,1] doesn't apply

> > here, because the outcome of an experiment is never a true real number

> > (most of which cannot even be written down !). There are always a

> > finite number of possibilities in the outcome of an experiment

> > (otherwise it couldn't be written onto a hard disk!).

>

> This does not really help. Let eps be the "extremely small probability"

> according to your proposal. Pick N >> - log eps / log 2, and

> run a series of N coin tosses. You get the result x_1 ... x_N, say.

> Although you really obtained exactly this result, the probability of

> obtaining it was only 2^{-N} << eps. Thus your proposal amounts to

> proving the impossibility of tossing coins more than a fairly small

> number of times

Maybe that "fairly small number of times" is in fact a very big

number, and in our universe there's not enough matter and time to do

all this tossing around!

A very small cutoff can save ALL of frequentist interpretations of

probability, because you are allowed to consider combined, independent

events (what's the probability of tossing 100 times a coin and finding

100 heads in a row AND seeing the moon go supernova etc...).

cheers,

Patrick.

Mar 31, 2004, 2:30:17 AM3/31/04

to

What is 'fundamental'? Stochastic processes are mathematically sound,

well-founded, and consistent, unlike current quantum field theory,

say. So they'd make a better foundation.

Chaotic processes are also idealizations, no less than stochastic

processes. And who can tell what is more basic? You can get one

from the other in suitable approximations...

Arnold Neumaier

Mar 31, 2004, 5:36:20 PM3/31/04

to

Patrick Van Esch wrote:

> Arnold Neumaier <Arnold....@univie.ac.at> wrote in message news:<406803CA...@univie.ac.at>...

>

>>But where exactly would you take it? If there is an eps such that

>>p<eps means 'it will not occur', what is the supremum of all such eps?

>>It would have to be a fundamental constant of nature.

>>

>>The nonexistence of such a constant implies that your proposal cannot

>>be right.

>>

>

>

> I agree that it should be a constant of this universe, if ever there

> was such a thing. It is not unthinkable that there IS such a constant

> (such as the inverse of the number of spacetime events, which could be

> finite).

> Arnold Neumaier <Arnold....@univie.ac.at> wrote in message news:<406803CA...@univie.ac.at>...

>

>>But where exactly would you take it? If there is an eps such that

>>p<eps means 'it will not occur', what is the supremum of all such eps?

>>It would have to be a fundamental constant of nature.

>>

>>The nonexistence of such a constant implies that your proposal cannot

>>be right.

>>

>

>

> I agree that it should be a constant of this universe, if ever there

> was such a thing. It is not unthinkable that there IS such a constant

> (such as the inverse of the number of spacetime events, which could be

> finite).

But the constant, to be meaningful in our real life, would have to be

not extremely small; and then it can be refuted easily.

> But I do realise how speculative that statement is.

> Nevertheless, without this vague idea, I cannot make any sense of what

> it experimentally means to have an event with a probability p.

For a single event, it means almost nothing.

For a large number of events, it means roughly the

relative frequency, but with a possibility of deviating to a not

precisely specified amount.

> Or

> better, why every time we do statistics on real data, it works out !

The sense it makes is the following: If you have a sound probabilistic

model of a multitude of independent events e_i with assigned

probability p you'd be surprised if the frequency of events is not

close to p within a small multiple of sqrt(p(1-p)/N). And you'd probably

rather try to explain away a rare occurence (a brick going upwards due

to fluctuations) by assuming a hidden, unobserved cause (someone throwing

it) rather than just accept it as something within your probabilistic

mode. The way probabilities are used in practice is always as rough guides

of what to expect, but not as statements with a 100% exact meaning.

I wrote a paper on surprise:

A. Neumaier,

Fuzzy modeling in terms of surprise,

Fuzzy Sets and Systems 135 (2003), 21-38.

http://www.mat.univie.ac.at/~neum/papers.html#fuzzy

that helps understand the fuzziness inherent in our concepts of reality.

>>>Your argument of drawing a real number out of [0,1] doesn't apply

>>>here, because the outcome of an experiment is never a true real number

>>>(most of which cannot even be written down !). There are always a

>>>finite number of possibilities in the outcome of an experiment

>>>(otherwise it couldn't be written onto a hard disk!).

>>

>>This does not really help. Let eps be the "extremely small probability"

>>according to your proposal. Pick N >> - log eps / log 2, and

>>run a series of N coin tosses. You get the result x_1 ... x_N, say.

>>Although you really obtained exactly this result, the probability of

>>obtaining it was only 2^{-N} << eps. Thus your proposal amounts to

>>proving the impossibility of tossing coins more than a fairly small

>>number of times

>

>

> Maybe that "fairly small number of times" is in fact a very big

> number, and in our universe there's not enough matter and time to do

> all this tossing around!

Oh, this is possible only if the universal eps is so tiny that

almost everything is possible - against the use you wanted to make

of it in real life! An ordinary person would take the eps to justify

their unconcious probabilistic models for assessing ordinary reality

quite high, for events that are not very repetitive probably at

1e-6 or so. (Even engineers who are responsible for the safety

of buildings, airplanes, etc.) This can only be justified with

being prepared to take the risk, but not with an objective cutoff.

Arnold Neumaier

Apr 1, 2004, 5:17:21 AM4/1/04

to

Arnold Neumaier <Arnold....@univie.ac.at> wrote in message news:<c4fh54$uur$1...@lfa222122.richmond.edu>...

>

> For a single event, it means almost nothing.

> For a large number of events, it means roughly the

> relative frequency, but with a possibility of deviating to a not

> precisely specified amount.

I know, and that's what you usually do, and the funny thing is that

the deviation is not very high ! I will call, when experimental

results follow in this sense the predictions of probability, a

"statistically correct" experiment.

But you realize the problems here:

first of all, all observation, even if it is 10000 times flipping a

coin, is a *single event* when taken as a whole, which has a high

probability when the event is "statistically correct" and a low

probability when it is not. We observe that experimental results of

this single event which are not statistically correct *never* occur.

This, to me, is a kind of miracle if there's not some kind of "law"

stating exactly this. Now I know the statistical physics explanation

of course: the high and low probabilities of these combined events

just reflect the fact that we deal with subsets of events with

different sizes: the subset of all sequences of 10000 heads and tails

which have an average around 5000 heads and 5000 tails is much bigger

than the subset with 10000 heads, which essentially contains just one

sequence. So if you hit one sequence "blindly" you'd probably hit one

of the biggest subsets. In fact, when you are in such a case, you

don't really need probability theory as such, it is just a matter of

*counting* equivalent points.

However, I have much more difficulties with quantum mechanical

probability predictions. After all, these are more fundamental

predictions because not the result of "picking blindly into a big set

of possiblities". So in order to make sense of the QM probability

predictions, we need a better understanding of exactly what is meant

by the frequentist interpretation of events with probability p. Now I

know that the frequentist interpretation is not the favourite one

amongst theoreticians, but I don't see how, as an experimentalist, you

can get around it.

cheers,

Patrick.

Apr 1, 2004, 10:15:23 AM4/1/04

to

On 2004-03-30, Patrick Van Esch <van...@ill.fr> wrote:

> The Bayesian interpretation of probabilities is a nice information

> theoretical construct of course, I'm not expert enough in it to see if

> the problem also exists there. But I have difficulties with people

> who deny the frequentist interpretation: after all, this is - to me -

> the only way to make a connection to experimental results ! How do

> you verify differential cross sections ? You do a number of

> experiments, and then you make the HISTOGRAM (counting the number of

> occurences) of the outcomes which you compare with your calculated

> probability density. That's nothing else but applying the frequentist

> interpretation, no ?

> The Bayesian interpretation of probabilities is a nice information

> theoretical construct of course, I'm not expert enough in it to see if

> the problem also exists there. But I have difficulties with people

> who deny the frequentist interpretation: after all, this is - to me -

> the only way to make a connection to experimental results ! How do

> you verify differential cross sections ? You do a number of

> experiments, and then you make the HISTOGRAM (counting the number of

> occurences) of the outcomes which you compare with your calculated

> probability density. That's nothing else but applying the frequentist

> interpretation, no ?

You can view it that way, but it comes nicely out of the Bayesian

interpretation too. Basically, as the number of samples goes up, the

probability of getting something markedly different than having the

right frequencies gets incredibly small.

--

Aaron Denney

-><-

Apr 1, 2004, 10:45:09 AM4/1/04

to

Patrick Van Esch wrote:

> We observe that experimental results of [[ 10000 coin flip example]]

> this single event which are not statistically correct *never* occur.

> This, to me, is a kind of miracle if there's not some kind of "law"

> stating exactly this. Now I know the statistical physics explanation

> of course: the high and low probabilities of these combined events

> just reflect the fact that we deal with subsets of events with

> different sizes: the subset of all sequences of 10000 heads and tails

> which have an average around 5000 heads and 5000 tails is much bigger

> than the subset with 10000 heads, which essentially contains just one

> sequence. So if you hit one sequence "blindly" you'd probably hit one

> of the biggest subsets. In fact, when you are in such a case, you

> don't really need probability theory as such, it is just a matter of

> *counting* equivalent points.

> However, I have much more difficulties with quantum mechanical

> probability predictions. After all, these are more fundamental

> predictions because not the result of "picking blindly into a big set

> of possiblities". So in order to make sense of the QM probability

> predictions, we need a better understanding of exactly what is meant

> by the frequentist interpretation of events with probability p. Now I

> know that the frequentist interpretation is not the favourite one

> amongst theoreticians, but I don't see how, as an experimentalist, you

> can get around it.

> We observe that experimental results of [[ 10000 coin flip example]]

> this single event which are not statistically correct *never* occur.

> This, to me, is a kind of miracle if there's not some kind of "law"

> stating exactly this. Now I know the statistical physics explanation

> of course: the high and low probabilities of these combined events

> just reflect the fact that we deal with subsets of events with

> different sizes: the subset of all sequences of 10000 heads and tails

> which have an average around 5000 heads and 5000 tails is much bigger

> than the subset with 10000 heads, which essentially contains just one

> sequence. So if you hit one sequence "blindly" you'd probably hit one

> of the biggest subsets. In fact, when you are in such a case, you

> don't really need probability theory as such, it is just a matter of

> *counting* equivalent points.

> However, I have much more difficulties with quantum mechanical

> probability predictions. After all, these are more fundamental

> predictions because not the result of "picking blindly into a big set

> of possiblities". So in order to make sense of the QM probability

> predictions, we need a better understanding of exactly what is meant

> by the frequentist interpretation of events with probability p. Now I

> know that the frequentist interpretation is not the favourite one

> amongst theoreticians, but I don't see how, as an experimentalist, you

> can get around it.

The "miracle" is not only the cardinality of contributing sets of events,

but also the *ergodicity*, which makes the statistics actually work in

practice.

Now, let's speculate (without selling our multi-souls to Multi-Devil...)

about a variant of many-world model of the Quantum Reality. Imagine that

there ARE effectively many worlds, each of them forming a fibre upon the

configuration substrate. When an electron may pass through a double slit,

in one subset of worlds it passes by one, in the other - through the other.

I am *NOT* speaking about the decoherence. No "splitting" takes place, the

fibres *are there*, and as in many other fibrous space you may choose your

fibres as you wish. Since this picture is embedded within a normal quantum

evolution picture, you may imagine also the "fusion"; you follow two different

fibres, but which finally end-up as one, since this is just drawing lines

in space, not physics.

And now, the dynamics in this fibrous space is *ERGODIC*. A kind of chaos in

multi-space... Following one fibre long enough to be able to repeat one

experiment (with identical preparation) many times, should give you the

distribution obtained from many fibres, from a "statistical ensemble" of them.

And you get a probabilistic model for the quantum reality. Of course, I didn't

really explain anything, I just shifted the focus from one set of words to

another. But such a "model" might rise its head again, when in some unspecified

future the experimentalists making very, very delicate measurements discover

that there are non-linear disturbances of the linear superposition principle.

I believe that one day the actual quantum theory will be replaced by something

else. Of course we won't get back to classical physics. If we discover some

non-linearities, then we will probably have to change our probabilistic/fre-

quentist or whatever interpretation, but let's wait...

Jerzy Karczmarczuk

Apr 4, 2004, 8:37:02 AM4/4/04

to

van...@ill.fr (Patrick Van Esch) wrote in message news:<c23e597b.04033...@posting.google.com>...

> Now I

> know that the frequentist interpretation is not the favourite one

> amongst theoreticians, but I don't see how, as an experimentalist, you

> can get around it.

>

>

> cheers,

> Patrick.

Actually, I think experimentalists use the Baysian approach. Usually

an experiment is undertaken with the expectation of some result. If

the results do not match this expectation, the equipment is tweaked

until the expected result is obtained. If this doesn't work either

the experiment is dropped or (rarely) some other explanation is found.

It is also true that physics experiments and games of chance are

deliberately constructed so that the frequentist model holds. Other

applications of the frequentist model, such as predictions of the

weather, are a looser application of statistics. In many sciences the

application of statistics falls in the "better than nothing" category.

What problem do theoretical physicists have with the frequentist

approach? I don't see how else QED could be interpreted. For

cosmology it is somewhat questionable.

Apr 4, 2004, 8:37:00 AM4/4/04

to

nos...@de-ster.demon.nl (J. J. Lodder) wrote in message news:<1gba5ru.1be...@de-ster.xs4all.nl>...

> All coins are quantum coins, for we live in a quantum word.

Well said.

> In practice it may be quite hard to say whether or not

> a coin throw may be considered to be 'classical'.

> Quantum mechanics may come in in the precise timing

> of the twitching of your fingers, on the molecular level,

> when flipping the coin.

>

Quantum mechanics comes into coins also in the impossibility to

fix/determine initial conditions.

It would be interesting to have estimates for the growth of initial

"quantum scale" indeterminacies in classical models of chaotic

physical systems.

Consider for example a set of macroscopic balls bouncing in a box. One

may assume that deviations grow by a factor 10 between bounces (this

is based on my experience as a billiard player) in a reasonable

position-momentum norm. If the above assumption is realistic the

uncertainty principle prevents macroscopically accurate* deterministic

forecasts spanning more than a few dozen bounces.

IV

* of the kind that's relevant in actual billiard games.

Apr 4, 2004, 8:37:04 AM4/4/04

to

"Arnold Neumaier" <Arnold....@univie.ac.at> wrote in message

news:c4fh54$uur$1...@lfa222122.richmond.edu...

> The sense it makes is the following: If you have a sound probabilistic

> model of a multitude of independent events e_i with assigned

> probability p you'd be surprised if the frequency of events is not

> close to p within a small multiple of sqrt(p(1-p)/N). And you'd probably

> rather try to explain away a rare occurence (a brick going upwards due

> to fluctuations) by assuming a hidden, unobserved cause (someone throwing

> it) rather than just accept it as something within your probabilistic

> mode. The way probabilities are used in practice is always as rough guides

> of what to expect, but not as statements with a 100% exact meaning.

> I wrote a paper on surprise:

> A. Neumaier,

> Fuzzy modeling in terms of surprise,

> Fuzzy Sets and Systems 135 (2003), 21-38.

> http://www.mat.univie.ac.at/~neum/papers.html#fuzzy

> that helps understand the fuzziness inherent in our concepts of reality.

This brings about an interesting possibility that the cutoff is anthropic.

Things that are statistically improbable (from the point of the theory we

are testing), even if they happen, are rejected. Conversely, if too many

improbable things happen, we reject the theory. So there is no correct or

incorrect theory (as long as it's self-consistent), only the currently

accepted one. Moreover, a theory once believed to be experimentally

confirmed (within a very good margin of error) might at some point be

rebuked by another set of identical experiments.

There is a very good example of this phenomenon--the evolution of the speed

of light since 1935. After the first publication of the Michelson

measurement, up till1947, all the measurement where lower than the currently

accepted value by more than the (admitted) experimental error (see diagram

at http://www.sigma-engineering.co.uk/light/lightindex.shtml). This was

probably caused by the experimenters rejecting the data points that were too

far from the then accepted Michelson's number. Notice that even here I'm not

considering the possibility that there was a 12-year long statistical

fluctuation ;-)

Apr 5, 2004, 3:02:06 PM4/5/04

to

Patrick Powers wrote:

> Actually, I think experimentalists use the Baysian approach. Usually

> an experiment is undertaken with the expectation of some result. If

> the results do not match this expectation, the equipment is tweaked

> until the expected result is obtained. If this doesn't work either

> the experiment is dropped or (rarely) some other explanation is found.

I've often worried that the vaunted accuracy of QED is illusory, that

is, data are used to "tune" the equipment.

-drl

Apr 6, 2004, 1:55:52 PM4/6/04

to

nos...@de-ster.demon.nl (J. J. Lodder) wrote in message news:<1gba5ru.1be...@de-ster.xs4all.nl>...

> All coins are quantum coins, for we live in a quantum word.

Well said.

> In practice it may be quite hard to say whether or not

> a coin throw may be considered to be 'classical'.

> Quantum mechanics may come in in the precise timing

> of the twitching of your fingers, on the molecular level,

> when flipping the coin.

>

Quantum mechanics kicks into coins also in the impossibility to

fix/determine initial conditions.

In a chaotic system an initial "quantum scale" indeterminacy will

quckly grow macroscopic, as highlighted in "Newtonian Chaos +

Heisenberg Uncertainty = macroscopic indeterminacy" by Barone, S.R.,

Kunhardt, E.E., Bentson, J., and Syljuasen, A., American Journal of

Physics, Vol 61, No. 5, May 1993.

Cheers,

IV

Apr 6, 2004, 1:57:27 PM4/6/04

to

Arnold Neumaier <Arnold....@univie.ac.at> writes:

>> r...@maths.tcd.ie wrote in message news:<c3njd0$295l$1...@lanczos.maths.tcd.ie>...

>>>For example, you'll find in many probability

>>>books and hear from the mouths of top probability theorists the

>>>claim that no process can produce random, uniformly distributed

>>>positive integers, but that processes can produce random uniformly

>>>distributed real numbers between zero and one (e.g. toss a fair

>>>coin exactly aleph_0 times to get the binary expansion).

>This has a very simple reason: There is no consistent definition of

>random, uniformly distributed positive integers, while there is

>one for random uniformly distributed real numbers between zero and one.

Please give the definition you claim exists.

>This is a purely mathematical statement independent of any

>interpretation!

Wow.

>And of course, when people say 'produce' they mean

>'produce in theory', or if they mean 'produce in practice' they

>have in mind that it is produced only approximately.

My point was that probability distributions and methods of

generating random numbers are not in one-to-one correspondance, and

I gave an example of a method of generating integers which had

no corresponding probability distribution. I too meant "produce

in theory", since obviously we can't use the axiom of choice in

real life, but if we want to understand what probability theory

is and isn't about (in theory), then we shouldn't make mistakes

on this fundamental point.

R.

Apr 6, 2004, 5:52:47 PM4/6/04

to

frisbie...@yahoo.com (Patrick Powers) writes:

>r...@maths.tcd.ie wrote in message news:<c3njd0$295l$1...@lanczos.maths.tcd.ie>...

>>

>> Right, so actually, the frequentist interpretation of probability

>> suffers from the same disease that the many-worlds interpretation

>> does, or at least the non-Bayesian one. In many worlds, the problem

>> is that there's no way to justify dismissing worlds with a small

>> quantum amplitude as being rare, and in the frequentist

>> version of probability theory, there's no way to justify dismissing

>> outcomes with small probability as being rare.

>>

>Quantum theory is a probabilistic theory and extremely unlikely events

>are not excluded, nor should they be. So this is a property of the

>theory, not the interpretation. It seems to me that an interpretation

>that excluded such events absolutely would be in error.

I'm not saying that they should be excluded; as a good Bayesian

I would merely say that the information available to me leads

me to expect that they won't happen, although it's not impossible.

>> The frequentist interpretation of probability suffers from worse

>> diseases as well. For example, you'll find in many probability

>> books and hear from the mouths of top probability theorists the

>> claim that no process can produce random, uniformly distributed

>> positive integers, but that processes can produce random uniformly

>> distributed real numbers between zero and one (e.g. toss a fair

>> coin exactly aleph_0 times to get the binary expansion).

>Yes these claims as stated are contradictory. I suspect that the

>definitions you are using are imprecise. The word "process" implies

>computability, that the process is finite. A real number is cleverly

>defined as a limit of a finite process. So a real number is

>computable in this sense, that it can be approximated as closely as

>one likes in finite time. The problem with your proof is that as the

>real number is computed the choice of cosets changes with each step so

>the process does not converge to an integer.

You are right; there is no convergence and there's no way to

actually compute such an integer or any approximation to it

in a finite number of operations. Modern mathematics, however,

allows us to deal with infinite sets without having to always

consider what can and can not be done in a finite number of

operations. The set of (not necessarily continuous) functions

from R to R has a cardinality greater than R itself, for example,

although this fact is of no relevance to finite creatures like

us. We don't *need* to define reals in terms of limits (for example,

we can define them in terms of Dedekind cuts).

So, rather than considering what's happening with the

cosets as being something which happens while the number is being

generated, I suppose that some acquaintance of mine can merely

give me a random number between 0 and 1, and then I convert

it into an integer. You, for example, might tell me 0.5 which

you can do in finite time, having generated it by your own

algorithm, which might be "just pick the middle number". If my

choice function is independent of your choice of number, then the

integer corresponding to 0.5 will be as random as the integer

corresponding to any other real number.

On the other hand, your point is well taken; generating reals

between 0 and 1 is itself impossible in practice.

>Using the axioms of choice and infinity then one can indeed choose a

>natural number at random. There are some rather strange consequences.

> It is then possible to prove that each number chosen in this way will

>be greater than all such previously chosen numbers with probability

>one. Let N be the greatest such number chosen so far. Then there are

>finitely many natural numbers less than or equal to N but infinitely

>many greater than N. So the next number chosen will be greater than N

>with probability one. Note that our ostensibly random sequence is

>strictly increasing with probability one.

If we consider infinite processes then the notion that the probabilities

one or zero mean anything goes out the window. Note that if I

tell you the third "random" integer first, then with probability

one the second is bigger than it, so with probability one the sequence

is not strictly increasing.

>This is not the only

>bizarre consequence of the axiom of choice: see the well-known

>Banach-Tarski sphere paradox. So I should think a physicist would do

>well to be wary of the axiom of choice as tending to produce

>non-physical results.

Indeed; these are more mathematical facts than physical ones.

Also, you don't need the axiom of choice to produce things like

Banach-Tarski - the set of complex numbers:

A={\sum a_n exp(i*n): a_n in N}, where N is the set of non-negative

integers can be broken into A=B disjoint union C, where B is A+1 and

C is exp(i)*A. Both B and C are exactly the same shape and size

as A, B being a translated version of A and C being a rotated version

of A, so A can be broken into two parts, each as "big" as A itself.

What I'm saying is that physicists don't need the axiom of choice

to get "unphysical" results. Infinite sets, which we use all the

time, are sufficient.

>The frequentist approach does not assume the axiom of choice and makes

>no use of transfinite mathematics or completed limits. If it did, the

>problems you mention would in fact arise.

Well, the problems I mentioned arise if we believe the axiom

of choice, the "experiment generating random numbers" version of

probability theory, and the idea that we can deal with infinite

sets (an axiom of infinity, eg "there exist infinite sets"), all

at the same time.

R.

Apr 6, 2004, 5:53:30 PM4/6/04

to

Bartosz Milewski wrote:

> "Arnold Neumaier" <Arnold....@univie.ac.at> wrote in message

> news:c4fh54$uur$1...@lfa222122.richmond.edu...

>

>>The sense it makes is the following: If you have a sound probabilistic

>>model of a multitude of independent events e_i with assigned

>>probability p you'd be surprised if the frequency of events is not

>>close to p within a small multiple of sqrt(p(1-p)/N). And you'd probably

>>rather try to explain away a rare occurence (a brick going upwards due

>>to fluctuations) by assuming a hidden, unobserved cause (someone throwing

>>it) rather than just accept it as something within your probabilistic

>>mode. The way probabilities are used in practice is always as rough guides

>>of what to expect, but not as statements with a 100% exact meaning.

>>I wrote a paper on surprise:

>> A. Neumaier,

>> Fuzzy modeling in terms of surprise,

>> Fuzzy Sets and Systems 135 (2003), 21-38.

>> http://www.mat.univie.ac.at/~neum/papers.html#fuzzy

>>that helps understand the fuzziness inherent in our concepts of reality.

>

>

> This brings about an interesting possibility that the cutoff is anthropic.

> "Arnold Neumaier" <Arnold....@univie.ac.at> wrote in message

> news:c4fh54$uur$1...@lfa222122.richmond.edu...

>

>>The sense it makes is the following: If you have a sound probabilistic

>>model of a multitude of independent events e_i with assigned

>>probability p you'd be surprised if the frequency of events is not

>>close to p within a small multiple of sqrt(p(1-p)/N). And you'd probably

>>rather try to explain away a rare occurence (a brick going upwards due

>>to fluctuations) by assuming a hidden, unobserved cause (someone throwing

>>it) rather than just accept it as something within your probabilistic

>>mode. The way probabilities are used in practice is always as rough guides

>>of what to expect, but not as statements with a 100% exact meaning.

>>I wrote a paper on surprise:

>> A. Neumaier,

>> Fuzzy modeling in terms of surprise,

>> Fuzzy Sets and Systems 135 (2003), 21-38.

>> http://www.mat.univie.ac.at/~neum/papers.html#fuzzy

>>that helps understand the fuzziness inherent in our concepts of reality.

>

>

> This brings about an interesting possibility that the cutoff is anthropic.

Not only anthropic, but subjective. Different people have different

views on the matter and are prepared to take different risks.

> Things that are statistically improbable (from the point of the theory we

> are testing), even if they happen, are rejected. Conversely, if too many

> improbable things happen, we reject the theory. So there is no correct or

> incorrect theory (as long as it's self-consistent), only the currently

> accepted one.

Positrons were observed before they were predicted by theory, but the

observers didn't believe the phenomenon was real. Rather than face

ridicule with a premature publication they ignored their evidence.

On the other hand, cold fusion had a different story...

We take a small probability p serious only if the associated phenomena

are repeatable frequently enough that an approximate frequentist

interpretation makes sense.

Arnold Neumaier

Apr 7, 2004, 6:45:05 AM4/7/04

to

In article <9511688f.04032...@posting.google.com>,

Patrick Powers <frisbie...@yahoo.com> wrote:

>Using the axioms of choice and infinity then one can indeed choose a

>natural number at random

>From context, let me add "with a uniform distribution" -- that is,

with all natural numbers equally probable.

Is this statement meant to be obvious? It's not at all clear to me

how the axiom of choice says anything about probabilities.

If it's not meant to be obvious, but is nonetheless true, can someone

point me to an appropriate place to read more on this?

-Ted

--

[E-mail me at na...@domain.edu, as opposed to na...@machine.domain.edu.]

Apr 8, 2004, 2:26:49 PM4/8/04

to

eb...@lfa221051.richmond.edu says...

>In article <9511688f.04032...@posting.google.com>,

>Patrick Powers <frisbie...@yahoo.com> wrote:

>

>>Using the axioms of choice and infinity then one can indeed choose a

>>natural number at random

>

>>From context, let me add "with a uniform distribution" -- that is,

>with all natural numbers equally probable.

>

>Is this statement meant to be obvious? It's not at all clear to me

>how the axiom of choice says anything about probabilities.

>

>If it's not meant to be obvious, but is nonetheless true, can someone

>point me to an appropriate place to read more on this?

I'm cross-posting to sci.math, because maybe a mathematician

has something to add. Patrick's point is not complicated

to prove, but it's hard to understand how to interpret it.

1. Pick an enumeration of all positive rational numbers

between 0 and 1. For example, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5,

3/5, 4/5, ... Let q_n be the nth rational number.

2. Define an equivalence relation on real numbers between

0 and 1: x ~~ y if and only if |x-y| is rational.

3. Using the axiom of choice, construct a set S by picking

one element out of every equivalence class.

4. Define S_n to be { x | |x - q_n| is in S }

Note that S_0 union S_1 union S_2 union ... = (0,1).

5. So here's how you generate a random nonnegative integer: Generate

a random real x in (0,1), and let your random integer be that n such

that x is an element of S_n.

There is no probability distribution on the possible outcomes of this

process, so it isn't a "uniform distribution on the integers" in a

measure-theoretic sense. But you can argue by symmetry that in some

sense every n is "equally likely" because each of the sets S_n are

identical, except for a translation.

--

Daryl McCullough

Ithaca, NY

Apr 8, 2004, 2:26:54 PM4/8/04

to

r...@maths.tcd.ie wrote:

> Arnold Neumaier <Arnold....@univie.ac.at> writes:

> Arnold Neumaier <Arnold....@univie.ac.at> writes:

>>There is no consistent definition of

>>random, uniformly distributed positive integers, while there is

>>one for random uniformly distributed real numbers between zero and one.

>

> Please give the definition you claim exists.

Just to know what kind of answer you'd be prepared to accept,

please let me know what you regard as the definition of random

binary numbers with equal probabilities of 0 and 1. Then I'll

be able to answer your more difficult question satisfactorily.

Arnold Neumaier

Apr 8, 2004, 6:38:28 PM4/8/04

to

eb...@lfa221051.richmond.edu wrote:

> In article <9511688f.04032...@posting.google.com>,

> Patrick Powers <frisbie...@yahoo.com> wrote:

>

>

>>Using the axioms of choice and infinity then one can indeed choose a

>>natural number at random

>

>

>>From context, let me add "with a uniform distribution" -- that is,

> with all natural numbers equally probable.

> In article <9511688f.04032...@posting.google.com>,

> Patrick Powers <frisbie...@yahoo.com> wrote:

>

>

>>Using the axioms of choice and infinity then one can indeed choose a

>>natural number at random

>

>

>>From context, let me add "with a uniform distribution" -- that is,

> with all natural numbers equally probable.

There is no uniform distribution on natural numbers.

There is no way to make formal sense of the statement

'all natural numbers equally probable'.

Thus this 'context' is logically meaningless.

The natural least informative distribution on natural numbers

is a Poisson distribution, but here one has to be at least informed

about the mean.

Arnold Neumaier

Apr 8, 2004, 6:40:54 PM4/8/04