Re: Fuzzy logic compared to probability

Herman Rubin (hrubin@b.stat.purdue.edu)
Fri, 8 Mar 1996 23:15:40 +0100


In article <DnsyEI.6u5@decan.com>, S. F. Thomas <sthomas@decan.com> wrote:
>Darren J Wilkinson (D.J.Wilkinson@durham.ac.uk) wrote:

.....................

>: Only a frequentist thinks of "repeated observations".

>I would have thought that Bayesians also were (in part) frequentist,
>since the likelihood function, which derives from frequentist
>probability models, is central to the updating of Bayesian prior
>belief.

This is a complete misunderstanding. Any attempt to come up with a
theory based upon action, and not based upon something as crude as
data description, requires that the assumptions come from the "mind".
Frequentism only comes in from the inference standpoint.

>: For a Bayesian, the
>: link is provided by consideration of symmetry and invariance, the
>: strongest form of which is given a name: "Exchangeability".

This is what I have been calling "rash Bayesian" behavior for decades.
There are two versions of invariance; changing the units, or changing
the presentation of the problem, would not change a Bayesian's results.
But if changing the units came up with a FORMALLY identical problem,
the results, in general, SHOULD be different. Symmetry likewise,
unless one's assumptions are symmetric.

...................

Subjective prior belief,
>no matter how sophisticated the mathematics, or how clever the
>trotting out of such concepts of "exchangeability", remains an
>artful dodge where the central problem of inference is concerned,
>which is, how to characterize what *the data* say about some unknown
>probability distribution of interest. Throwing in a statistician's --
>or a "user's" -- prior belief into the mix, is still, in my opinion,
>sidestepping the real question. And arguing for the *necessity*
>of so doing, not to mention the *goodness* of so doing, is the
>Bayesian equivalent of turning a bug into a feature.

You and too many others are overly concerned with belief. If, instead,
one starts with the idea of self-consistent behavior, which is much
weaker than one would think, it is still strong enough to be only
compatible with a weighted sum of the probabilities of outcomes in
the various states of nature, the weights depending on the action.
If these weights are called the product of loss and prior, this gets
the more customary formulation. There is no operational way of
separating loss from prior.

...................

>: The other is that such
>: understandings do not have to be fundamentally subjective. Frequentist
>: statisticians have been trying to be "objective" for decades, and the
>: literature is littered with examples of it's abject failure. I will end
> ^^^^^^^^^^^^^^
>: this post with a quote by a man who understood uncerainty better than
>: anyone had ever done before....

>You make assertions, or rather dogmatic statements of Bayesian
>doctrine. But you do not make rational arguments to which one could
>respond. Surely, not even a Bayesian would deny that observing
>heads or tails on the toss of a coin is essentially an "objective"
>procedure, whatever one's prior belief might be as to which
>might turn up?

The OBSERVATION is objective. The action to take should not be the same
for all observers. If it were, Michelson and Morley would have come up
with special relativity as soon as they made their observations. Newton
would have come up with the wave theory of light. Archimedes and Euclid
would have used variables. The structure of "elementary particles" would
have been deduced years before.

..................

-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu	 Phone: (317)494-6054	FAX: (317)494-0558