Re: probability (was Re: EEQT)
From: Arnold Neumaier (Arnold.Neumaier_at_univie.ac.at)
Date: 09/04/04
- Next message: Arnold Neumaier: "Re: Quantum-Classical Boundary"
- Previous message: grelbr: "Re: Numerical Integration of the Time-Dependent???"
- In reply to: Aaron Denney: "Re: probability (was Re: EEQT)"
- Messages sorted by: [ date ] [ thread ]
Date: Sat, 4 Sep 2004 07:04:52 +0000 (UTC)
Aaron Denney wrote:
> On 2004-08-26, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
>
>>Aaron Denney wrote:
>>
>>>On 2004-08-24, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
>>>
>>>>Aaron Denney wrote:
>>>>
>>>>>On 2004-08-19, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
>>>>>
>>>>>>Of course, in practice, we approximate real coins by 'fair coins'
>>>>>>defined through an infinite ensemble, since the latter is tractable in
>>>>>>any detail desired.
>>>>>
>>>>>Why do you need an infinite ensemble? A fair coin can be modeled just
>>>>>fine with an ensemble of size two.
>>>>
>>>>How do you toss a fair coin twice within an ensemble of size two?
>>>>You cannot get infinitely many independent trials with a finite sigma
>>>>algebra.
>>>
>>>You either re-use the ensemble (drawing with replacement),
>>
>>This is not a formally valid procedure; there is no way to tell
>>whether the reuse is independent. To give independence a mathematical
>>meaning one indeed needs to take your second choice:
>
> I really don't understand your objection here.
The concept of independent random variables is vacuous in a sigma algebra
over a sample space of size 2, since there is essentially only one
random variable.
>>>or you tensor it with itself (N - 1) times to describe the case of
>>>a coin tossed N times, with 2^N possibilities. The throws are then
>>>independent by construction.
>>
>>Yes. And to get infinitely many independent trials you neet to take
>>the tensor product of infinitely many copies, and the resulting sigma
>>algebra is no longer finite, as claimed.
>
> Do you ever need to throw a coin infinitely many times? You can't throw
> a real coin infinitely many times -- even disregarding the time it would
> take, it would wear away to nothing far before that.
In practice, you can have only a finite sample. But probability theory
needs infinite series of independent or nearly independent trials to make
assertions such as the law of large numbers.
> To model any number of fair tosses, it requires only a finite algebra.
> (terminology nit-pick: We're not interested in modelling "the fair
> coin". A coin can't be "fair", or "unfair". A coin toss, on the other
> hand, may or may not be, depending on the initial conditions implicitly
> assumed. Yes, I've been sloppy and used the former phrasing.)
We _model_ a real coin by a fair coin, which can be tossed arbitrarily
often, getting independent realizations. This makes it possible to apply
the machinery of probability to statistics, by pretending that large
finite numbers are a good approximation to infinity, and that random
numbers are a sensible way of representing randomly looking results in a
cleean, conceptual way.
But real coins _never_ produce random binary sequences (which are
sequences of random numbers = sequences of measurable functions,
not sequences of zeros and ones) but only finite sequences of actual
zeros and ones (or heads and tails) that look more or less random.
The way randomness is applied to a finite number of facts (and the number
of all facts known on earth is finite) is always in a somewhat loose way.
There is no way out of the inherent ambiguity in the relation between
concepts in the platonic realm and their shadows in relality, or between
real facts and their conceptual shadows in platonic reality.
Arnold Neumaier
- Next message: Arnold Neumaier: "Re: Quantum-Classical Boundary"
- Previous message: grelbr: "Re: Numerical Integration of the Time-Dependent???"
- In reply to: Aaron Denney: "Re: probability (was Re: EEQT)"
- Messages sorted by: [ date ] [ thread ]
Relevant Pages
|
