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I just posted this to sci.stat.edu. With apologies to those who would

see it twice, I post it again, this time cross-posted to

comp.ai.fuzzy, where it may also be of interest.

"Neville X. Elliven" wrote:

*>
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*> R. Jones wrote:
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*>
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*> >Can anyone refer me to a good book on the foundations of statistics?
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*> >I want to know of the limitations, assumptions, and philosophy
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*> >behind statistics.
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*>
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*> "Probability, Statistics, and Truth" by Richard von Mises is available
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*> in paperback [ISBN 0-486-24214-5] and might be just what you seek.
*

May I suggest my own "Fuzziness and Probability" (ACG Press, 1995).

In attempting to reconcile the competing paradigmatic claims to

representing uncertainty, of fuzzy set theory (FST) on the one hand,

and probability/statistical inference theory (PST) on the other, I

was driven to look deeply into the foundations, not only of these

two, but also of measurement theory, deductive and inductive logic,

decision analysis, and the relevant aspects of semantics. I also

found it necessary to be clear as to the notion of what constitutes a

model, and logically prior to that, what constitutes a phenomenon,

which competing models seek in some way to represent. I think I have

succeeded, not only in reconciling the competing claims of FST and

PST, but also in finding the extended likelihood calculus which

eluded Fisher, and the generations of statisticians since. Likelihood

theory thus far has been considered inadequate because simple

maximization rules of maginalization and set evaluation fail in

significant cases, which may in part have temptingly led Bayesians to

substitute a probabilistic model, now necessarily subjectivist, for

what is in actuality a possibilistic sort of uncertainty.

Classicists, quite rightly, have never accepted this insistent

Bayesian subjectivism, while Bayesians, quite understandably, have

been impatient with the cautious, indirect characterizations of

statistical uncertainty that are the hallmark of classical

(Neyman-Pearson) statistical method. An extended likelihood calculus

which is as easy of manipulation as the probability calculus, but

without the injection of subjective priors, seems to me to offer a

solution to the disagreements that beset the foundations of

statistical inference. At any rate, the original poster may want to

take a look see. Be all that as it may, I would also commend to the

original poster to the following two sources, which I found to be

very helpful when I was asking the sorts of questions which the

original poster now poses:

1) Sir Ronald A. Fisher. (1951). Statistical Methods and Scientific

Inference. Collier MacMillan, 1973 (third edition).

2) V.P. Godambe and D.A. Sprott (eds.). (1971). Foundations of

Statistical Inference: A Symposium. Toronto, Montreal: Holt, Rinehart

and Winston.

There are many other worthwhile references, but these two helped me

enormously in framing the core issues. The latter was especially

useful for the informal commentaries and rejoinders which saw

respective champions of the three main schools of thought --

classical, Bayesian, and likelihood -- going at each other in

vigorous debate.

*> >A discussion of how the quantum world may have
*

*> >different laws of statistics might be a plus.
*

*>
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*> The statistical portion of statistical mechanics is fairly simple, and
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*> no different conceptually from other statistics.
*

But from the standpoint of one whose interest is in the

quantum-theoretic application domain, there is a very real question

of where fuzziness ends, and probability begins. I remember reading

Penrose's "The Emperor's New Mind", and thinking -- idly, it's not my

field and I haven't tried to follow up -- that at least some of the

uncertainty in the quantum world is of the fuzzy rather than

probabilistic sort. The original poster is certainly well-advised to

explore the foundations of uncertainty, period, as distinct from

purely statistical uncertainty.

Hope this is of some help.

Regards,

S. F. Thomas

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