[extropy-chat] what is probability?
gts
gts_2000 at yahoo.com
Tue Jan 16 14:59:42 UTC 2007
On Mon, 15 Jan 2007 13:12:46 -0500, Russell Wallace wrote..
>> If that's still not enough for a reliable answer, you go out and gather
>> data by polling substantial numbers of people, and use the result of
>> that as your prior for further investigation.
Revisiting this...
Our bayesian-minded pollster wants to estimate the % of democrats among
voters in a given district. For whatever reason he believes (and we
stipulate) that he has no reliable prior data for use in bayes' theorem.
Ordinarily this would be a situation in which the researcher would invoke
the principle of indifference to obtain a prior, but we have doubts about
the legitimacy of the PI. He follows our recommendation that he ignore the
PI and instead "go out and gather data by polling substantial numbers of
people, and use the result of that as your prior for further
investigation."
But "polling substantial numbers of people" to obtain a probability is
exactly what our researcher had intended in the first place! So now his
job is already done, but he's done it with frequentist rather than
bayesian methods.
By denying him use of the principle of indifference we have converted him
from a logical bayesian into an objective frequentist (possibly while
kicking and screaming).
This is how the PI is integral to certain logical theories of probability.
If the PI is not a valid logical principle then the logical
interpretations of probability go down with it. Bayes' theorem survives of
course, but not in its *logical* implementation.
I think that is not such a bad thing, necessarily. It's quite a stretch I
think to assume as do the logical theorists that probability theory is a
branch of formal logic. Normally a conclusion is entailed by the premises,
and certain if the premises are true, but such is obviously not the case
when the premises are statistical data.
-gts
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