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| Neil B.... |
 Posted: Mon Apr 06, 2009 8:33 pm |
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Guest
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There are big problems talking clearly about randomness, in particular
how to rate the truthfulness of statements about probability. I have
long wondered how to treat the truth value of a statement like "70%
chance of rain today." How can we rate the truth value of such
statements? Neither raining nor not raining can show a single such
statement either true or false! Do such statements need to partake of a
"collective" truth value? Can we say, if we gather 1,000 such
predictions from a given forecaster and it rained only 40% of all those
times, the statements are collectively "not very true" etc? But of
course, what rightly defines the "collection" of note? What if we get an
impression the statements were less true in the beginning of a person's
career, we can't even define a stable pattern to characterize. (That's a
problem with probability in general. Talk of indefinite runs under "a
given probability" become tricky if the probability itself is subject to
change.) |
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| Chetan... |
| Posted: Mon Apr 06, 2009 10:21 pm |
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"Neil B." <neil_delver at (no spam) caloricmail.com> writes:
[quote:e9c259b6f7]There are big problems talking clearly about randomness, in particular
how to rate the truthfulness of statements about probability. I have
long wondered how to treat the truth value of a statement like "70%
chance of rain today." How can we rate the truth value of such
statements? Neither raining nor not raining can show a single such
statement either true or false! Do such statements need to partake of a
"collective" truth value? Can we say, if we gather 1,000 such
predictions from a given forecaster and it rained only 40% of all those
times, the statements are collectively "not very true" etc? But of
course, what rightly defines the "collection" of note? What if we get an
impression the statements were less true in the beginning of a person's
career, we can't even define a stable pattern to characterize. (That's a
problem with probability in general. Talk of indefinite runs under "a
given probability" become tricky if the probability itself is subject to
change.)
[/quote:e9c259b6f7]
All statements of probabilities imply averaging over a "sufficiently
large" sample space under similar conditions and therefore a single
event says nothing about the validity of the statement. What is
"sufficiently large" depends on the randomness of the event in
question. This will hold when the factors that affect the outcome are
random. In the case of person learning from experience, the learning
itself may affect the outcome and it is not a random factor, but has a
definite bias. From a large sample set, one can determine the
distribution and then use the result for testing a hypothesis about
one particular factor affecting the outcome.
The original statement, for example, amounts to saying that when
conditions are like today, the statement that "it would rain today" is
likely to be correct 70% of the times. However, what are the
conditions considered is not specified.
--
Chetan |
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| Herman Rubin... |
| Posted: Tue Apr 07, 2009 1:47 pm |
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In article <u1vs5tag8.fsf at (no spam) myhost.sbcglobal.net>,
Chetan <chetan.xspam at (no spam) xspam.sbcglobal.net> wrote:
[quote:3e0cc18edb]"Neil B." <neil_delver at (no spam) caloricmail.com> writes:
There are big problems talking clearly about randomness, in particular
how to rate the truthfulness of statements about probability. I have
long wondered how to treat the truth value of a statement like "70%
chance of rain today." How can we rate the truth value of such
statements? Neither raining nor not raining can show a single such
statement either true or false!
[/quote:3e0cc18edb]
One needs to be more careful than you are. A linear scale,
such as probability, is not a truth value but a measure of
truth value. The truth value system is the Boolean algebra.
Do such statements need to partake of a
[quote:3e0cc18edb]"collective" truth value? Can we say, if we gather 1,000 such
predictions from a given forecaster and it rained only 40% of all those
times, the statements are collectively "not very true" etc? But of
course, what rightly defines the "collection" of note? What if we get an
impression the statements were less true in the beginning of a person's
career, we can't even define a stable pattern to characterize. (That's a
problem with probability in general. Talk of indefinite runs under "a
given probability" become tricky if the probability itself is subject to
change.)
[/quote:3e0cc18edb]
There are methods of evaluating probability forecasts; they
are not that simple, but Bayesian methods exist.
[quote:3e0cc18edb]All statements of probabilities imply averaging over a "sufficiently
large" sample space under similar conditions and therefore a single
event says nothing about the validity of the statement.
[/quote:3e0cc18edb]
On the contrary, probability has to begin with events which
cannot be repeated. The law of large numbers ties relative
frequency to probability, and the central limit theorem
shows that one cannot estimate probability well by relative
freqyency.
What is
[quote:3e0cc18edb]"sufficiently large" depends on the randomness of the event in
question. This will hold when the factors that affect the outcome are
random. In the case of person learning from experience, the learning
itself may affect the outcome and it is not a random factor, but has a
definite bias. From a large sample set, one can determine the
distribution and then use the result for testing a hypothesis about
one particular factor affecting the outcome.
[/quote:3e0cc18edb]
This is a simplistic view, and does not lead to understanding.
The axioms of probability tell us what one can and what one
cannot do with probability models. Limiting things to
relative frequencies of independent events under "identical"
conditions makes it too weak.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin at (no spam) stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 |
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| Chetan... |
| Posted: Tue Apr 07, 2009 3:56 pm |
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Guest
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hrubin at (no spam) odds.stat.purdue.edu (Herman Rubin) writes:
[quote:4137c8f841]In article <u1vs5tag8.fsf at (no spam) myhost.sbcglobal.net>,
Chetan <chetan.xspam at (no spam) xspam.sbcglobal.net> wrote:
"Neil B." <neil_delver at (no spam) caloricmail.com> writes:
There are big problems talking clearly about randomness, in particular
how to rate the truthfulness of statements about probability. I have
long wondered how to treat the truth value of a statement like "70%
chance of rain today." How can we rate the truth value of such
statements? Neither raining nor not raining can show a single such
statement either true or false!
One needs to be more careful than you are. A linear scale,
such as probability, is not a truth value but a measure of
truth value. The truth value system is the Boolean algebra.
Do such statements need to partake of a
"collective" truth value? Can we say, if we gather 1,000 such
predictions from a given forecaster and it rained only 40% of all those
times, the statements are collectively "not very true" etc? But of
course, what rightly defines the "collection" of note? What if we get an
impression the statements were less true in the beginning of a person's
career, we can't even define a stable pattern to characterize. (That's a
problem with probability in general. Talk of indefinite runs under "a
given probability" become tricky if the probability itself is subject to
change.)
There are methods of evaluating probability forecasts; they
are not that simple, but Bayesian methods exist.
All statements of probabilities imply averaging over a "sufficiently
large" sample space under similar conditions and therefore a single
event says nothing about the validity of the statement.
On the contrary, probability has to begin with events which
cannot be repeated. The law of large numbers ties relative
frequency to probability, and the central limit theorem
shows that one cannot estimate probability well by relative
freqyency.
[/quote:4137c8f841]
Probabilistic events cannot be repeated "exactly". However, the
sample selection has to be random and free of bias to make the results
applicable.
[quote:4137c8f841] What is
"sufficiently large" depends on the randomness of the event in
question. This will hold when the factors that affect the outcome are
random. In the case of person learning from experience, the learning
itself may affect the outcome and it is not a random factor, but has a
definite bias. From a large sample set, one can determine the
distribution and then use the result for testing a hypothesis about
one particular factor affecting the outcome.
This is a simplistic view, and does not lead to understanding.
[/quote:4137c8f841]
The question _was_ simplistic.
[quote:4137c8f841]The axioms of probability tell us what one can and what one
cannot do with probability models. Limiting things to
relative frequencies of independent events under "identical"
conditions makes it too weak.
[/quote:4137c8f841]
That is what the question was about.
--
Chetan |
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| Peter11... |
| Posted: Sat Apr 18, 2009 4:43 pm |
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Guest
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Herman Rubin ha scritto:
[quote:e91d342c0b]
There are methods of evaluating probability forecasts; they
are not that simple, but Bayesian methods exist.
All statements of probabilities imply averaging over a "sufficiently
large" sample space under similar conditions and therefore a single
event says nothing about the validity of the statement.
On the contrary, probability has to begin with events which
cannot be repeated. The law of large numbers ties relative
frequency to probability, and the central limit theorem
shows that one cannot estimate probability well by relative
freqyency.
Nothing change...You start with a subjective distribution and[/quote:e91d342c0b]
parameters, and change it on the base of the experience.
A doctor can say that 80% of patient treated affected by pathology A
survive after a surgical treatment, but anything about the destiny of
the patient in front of him...
[quote:e91d342c0b]
W
The axioms of probability tell us what one can and what one
cannot do with probability models. Limiting things to
relative frequencies of independent events under "identical"
conditions makes it too weak.
[/quote:e91d342c0b]
The axioms tell us that p. of omega is one, that p. is a measure in the
segment [0,1] that transformation is closed with respect, at least, a
numerable union of events. Using carefully the smallest sigma algebra... |
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