Re: Isn't the Frequentist a subset or special case of Bayesianist?
- From: "Reef Fish" <Large_Nassau_Grouper@xxxxxxxxx>
- Date: 5 Jul 2005 08:22:32 -0700
vontressms@xxxxxx wrote:
> dawenliu@xxxxxxxxx wrote:
> > My shallow understanding of the Frequentist is that he puts a uniform
> > prior. If so, isn't this just a special case of the Bayesian approach,
> > where the prior can be anything, including a uniform distribution?
> >
> > Thanks
>
> There is a fundamental difference between Bayesian and frequentists
> philosophies. Both share the belief that a random variable may be
> described by a mathematical probability model which includes some
> unknown parameters. Bayesians accept the belief that these parameters
> are also random variables and have a probability distribution.
> Frequentists accept the belief that the parameters are unknown
> constants.
>
> This difference is subtle, but profound. The difference lies in a
> disagreement about what one may "observe" in a random experiment. In
> both cases, the parameters are unknown. Bayesians believe that a
> subjective experiment may be performed to divine the values of a
> parameter. I use the word "divine" somewhat fecitiously. Bayesians
> accept the idea that the distribution of this random variable called a
> parameter may be expressed in a subjective probability distribution.
> Subjective probability is different from frequentist probability since
> frequentist probability derives from repeated observations of a
> physically measurable observation. Subjective probability must be
> constructed from belief alone.
An excellent exposition and summary!
< snip >
> Most arguments about which philosophy is superior center around the
> willingess and benifits of accepting subjectivity into an emperical
> endevor.
>
> Mark
Here's an example L.J.Savage used to illustrate the use of "prior"
or "subjective" opinion in an estimation problem of the unknown p
of a binomial r.v.
It's almost NEVER the case that you don't have ANY idea where p
might be. Which is what a Frequentist assumes, and also a naive
Bayesian might use the U(0,1) prior (a member of the conjugate
familiar of priors for that distribution).
Savage let p be the probability that a person can correctly
distinguish a piece of music by Hayden and Handel.
Savage argues, and I think convincingly, that you would not
treat the evidence of the result of 10 binomial trials the
same way (whatever the number of successes is), if one of the
subjects is a well-known musicologist; and the other is a
drunk at the corner of the street paid to guess on each trial.
But that's what Frequentist so.
Not only that, the SAME experiment tried millions of times
under very similar conditions, the Frequentest ignores ALL
of them, and starts afresh with the same small number of trials
as if NOTHING is known about the unknown p.
Only recently has the word "meta analysis" enter into the
statistical jargon as a heuristic way of combining the evidence
from different experiments on the same problem.
A subjective Bayesian would be completely consistent in treating
a sample of ANY size -- each DATUM (an observation of size 1)
has the SAME impact on how it chages the PRIOR distribution,
whether that piece of datum is analyzed one at a time, 4 at a
time, or 100,000 all at once. The end result is Exactly the
same!
Moreover, a Bayesian's prior may be drastically wrong, but it
CANNOT remain very wrong for long, because the observations
necessarily shift the prior distribution to where it belongs
for the next experiment, the current posterior distriburion.
C'est la difference!
-- Bob.
.
- References:
- Isn't the Frequentist a subset or special case of Bayesianist?
- From: dawenliu
- Re: Isn't the Frequentist a subset or special case of Bayesianist?
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- Isn't the Frequentist a subset or special case of Bayesianist?
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