Re: Need Good book on foundations of statistics

From: Sidney Thomas (sf.thomas@verizon.net)
Date: Sat Jun 23 2001 - 07:51:28 MET DST

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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:
    >
    > R. Jones wrote:
    >
    > >Can anyone refer me to a good book on the foundations of statistics?
    > >I want to know of the limitations, assumptions, and philosophy
    > >behind statistics.
    >
    > "Probability, Statistics, and Truth" by Richard von Mises is available
    > 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.
    >
    > The statistical portion of statistical mechanics is fairly simple, and
    > 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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