In article <[EMAIL PROTECTED]>,
Dennis Roberts <[EMAIL PROTECTED]> wrote:
>in reference to a CI, the critical issue is CAN it be said that ... in the
>long run, there is a certain probability of producing CIs (using some CI
>construction procedure) that ... contain the parameter value ... that is,
>how FREQUENTLY we expect the CIs to contain the true value ... well, yes we can
>
>THAT is the important idea and, i think that if we try (for the sake of
>edification of the intro student)to defend it or reject it according to
>being proper bayesian/frequentist or improper ... is totally irrelevant to
>the basic concept
THAT is indeed the important idea for understanding what a frequentist
C. I. really is. Unfortunately, it is NOT the property that is important
to anyone who is actually using a confidence interval, who OF COURSE is
interested in whether the particular confidence interval they obtained
contains the true parameter value. I emphasize the OF COURSE because it
seems that some frequentists have managed to contort their thinking to
the point where they actually think that the long run coverage probability
of the C. I. is what users are interesed in, in defiance of all common
sense. More commonly, though, the tendency is to just give the Bayesian
interpretation, even though it is not justified.
This is not just an academic point. If you tell someone that the 95% C. I.
obtained has the Bayesian interpretation, when it isn't actually the result
of a Bayesian procedure, they may well decide that even though they had
previously thought that the parameter value was outside this interval, they
must have been wrong, since the statistician says there's a 95% chance the
parameter is in the C. I. This is all wrong. There are two more-or-less
right approaches, which are:
1) Use a frequentist C. I., while understanding that the parameter does
NOT necessarily have a 95% chance of being in the interval you obtained.
You have to informally weigh in your mind whether it is more likely that
the parameter is inside the interval, or that this is one of the 5%
of the intervals that don't contain the true parameter value. There's
no mathematical justification for deciding either way.
2) Use a Bayesian procedure instead, which will of course include specification
of a prior distribution for the parameter. Then you can find an interval
for which you can indeed say that the parameter has a 95% chance of lying
inside. (Or you might just look at the whole posterior distribution.)
If (1) sounds like a rather convoluted way of getting to a final situation in
which you make a subjective judgement, then maybe you would prefer (2), in which
the subjectivity is explict in the prior distribution.
Radford Neal
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Radford M. Neal [EMAIL PROTECTED]
Dept. of Statistics and Dept. of Computer Science [EMAIL PROTECTED]
University of Toronto http://www.cs.utoronto.ca/~radford
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