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From: Jesse Mazer <lasermazer.domain.name.hidden>

Date: Tue, 20 Jan 2004 23:38:30 -0500

Kory Heath wrote:

*>
*

*>At 1/19/04, Hal Finney wrote:
*

*>>However, here is an alternate formulation of my argument which seems to
*

*>>be roughly equivalent and which avoids this objection: create a random
*

*>>program tape by flipping a coin for each bit. Now the probability that
*

*>>you created the first program above is 1/2^100, and for the second,
*

*>>1/2^120, so the first program is 2^20 times more probable than the second.
*

*>
*

*>That's an interesting idea, but I don't know what to make of it. All it
*

*>does is create a conflict of intuition which I don't know how to resolve.
*

*>On the one hand, the following argument seems to make sense: consider an
*

*>infinite sequence of random bits. The probability that the sequence begins
*

*>with "1" is .5. The probability that it begins with "01" is .25. Therefore,
*

*>in the uncountably infinite set of all possible infinite bit-strings, those
*

*>that begin with "1" are twice as common as those that begin with "01".
*

*>However, this is in direct conflict with the intuition which says that,
*

*>since there are uncountably many infinite bit-strings that begin with "1",
*

*>and uncountably many that begin with "01", the two types of strings are
*

*>equally as common. How can we resolve this conflict?
*

*>
*

*>-- Kory
*

I haven't studied measure theory, but from reading definitions and seeing

discussions my understanding is that it's about functions that assign real

numbers to collections of subsets (defined by 'sigma algebras') of infinite

sets. As applied to probability theory, it allows you to define a notion of

probability on a set with an infinite number of members. Again, this would

involve assigning probabilities to *subsets* of this infinite set, not to

every member of the infinite set--for example, if you are dealing with the

set of real numbers between 0 and 1, then although each individual real

number could not have a finite probability (since this would not be

compatible with the idea that the total probability must be 1), perhaps each

finite nonzero interval (say, 0.5 - 0.8) would have a finite probability. In

a similar way, if you were looking at the set of all possible infinite

bit-strings, although each individual string might not get a probability,

you might have a measure that can tell you the probability of getting a

member of the subset "strings beginning with 1" vs. the probability of

getting a member of the subset "strings beginning with 01". Some references

on measure theory that may be helpful:

http://en2.wikipedia.org/wiki/Measure_theory

http://en2.wikipedia.org/wiki/Sigma_algebra

http://en2.wikipedia.org/wiki/Probability_axioms

http://mathworld.wolfram.com/Measure.html

http://mathworld.wolfram.com/ProbabilityMeasure.html

Jesse Mazer

_________________________________________________________________

Learn how to choose, serve, and enjoy wine at Wine -AT_SYMBOL- MSN.

http://wine.msn.com/

Received on Tue Jan 20 2004 - 23:41:32 PST

Date: Tue, 20 Jan 2004 23:38:30 -0500

Kory Heath wrote:

I haven't studied measure theory, but from reading definitions and seeing

discussions my understanding is that it's about functions that assign real

numbers to collections of subsets (defined by 'sigma algebras') of infinite

sets. As applied to probability theory, it allows you to define a notion of

probability on a set with an infinite number of members. Again, this would

involve assigning probabilities to *subsets* of this infinite set, not to

every member of the infinite set--for example, if you are dealing with the

set of real numbers between 0 and 1, then although each individual real

number could not have a finite probability (since this would not be

compatible with the idea that the total probability must be 1), perhaps each

finite nonzero interval (say, 0.5 - 0.8) would have a finite probability. In

a similar way, if you were looking at the set of all possible infinite

bit-strings, although each individual string might not get a probability,

you might have a measure that can tell you the probability of getting a

member of the subset "strings beginning with 1" vs. the probability of

getting a member of the subset "strings beginning with 01". Some references

on measure theory that may be helpful:

http://en2.wikipedia.org/wiki/Measure_theory

http://en2.wikipedia.org/wiki/Sigma_algebra

http://en2.wikipedia.org/wiki/Probability_axioms

http://mathworld.wolfram.com/Measure.html

http://mathworld.wolfram.com/ProbabilityMeasure.html

Jesse Mazer

_________________________________________________________________

Learn how to choose, serve, and enjoy wine at Wine -AT_SYMBOL- MSN.

http://wine.msn.com/

Received on Tue Jan 20 2004 - 23:41:32 PST

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