Moreover, it seems clear to me at least, that classical probability theory
is not the omnipotent calculus it is often portrayed to be. For instance,
in risk analysis, where incomplete knowledge is the rule, one generally
cannot get any answer at all without resorting to a subjectivist
interpretation of probability, maximum entropy criteria, empirically
unjustified assumptions of independence, or a combination of these and
other strategies that are divorced still further from what can be
justified by appeal to empirical information.
Here is a simple example. Suppose I tell you that A and B are numerical
inputs about whose values I am uncertain. I only know that the value or
values of A must be between 0.3 and 0.5, and the value(s) of B must be
between 0.2 and 0.4. What can be said about the sum A+B?
Because I know nothing about A or B except their ranges, many Bayesians
and most maximum entropy heads will use uniform distributions to model the
two inputs and (maybe) some assumption of independence to compute a
triangular distribution (0.5, 0.7, 0.9). This answer surely confounds
ignorance (imprecision) with randomness (variability). Of course, some
analysts wisely refuse to answer the question and produce no answer at
all, although this is hardly a tenable strategy in general.
As interval probability or robust Bayesian analysis or simple interval
analysis reveals, the answer is the interval [0.5, 0.9]. In my view, this
is the only correct answer from the point of view of a risk analysis who
cannot use subjectivist interpretations but must rely on a pure
frequentist interpretation. The interval corresponds to the *set* of
probability distributions whose supports are on [0.5, 0.9], but not to any
particular distribution from that set. It is the answer that is produced
natively by fuzzy arithmetic (which generalizes interval analysis) but
probability theory actually requires a rather sophisticated analysis to
arrive at the correct conclusion.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Scott Ferson Telephone 516-751-4350
Applied Biomathematics Facsimile 516-751-3435
100 North Country Road Email risk@life.bio.sunysb.edu
Setauket, New York 11733 USA Web http://gramercy.ios.com/~ramas
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On Fri, 24 May 1996, S. F. Thomas wrote:
> Barry O'Sullivan (osullb@cs.ucc.ie) wrote:
>
>
> : I think the easiest way to reconcile the differences between fuzzy
> : membership and probability is as follows. Probability is based on
> : randomness. Probability theory regards imprecision to be equated with
> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> : randomness.
> ^^^^^^^^^^
> This is a common misconception. It is not so.
>
> : On the otherhand, fuzziness is the type of imprecision
> : encountered when there does not exist a sharp transition between \
> : membership and non-membership.
>
> Yes. Now consider the set of Bernoulli parameters which explain
> the outcome on a single toss of say a thumb-tack for which we
> wish to estimate the probability of it landing top down on a
> single toss. Is there a sharp transition from membership to
> non-membership for such a set? If you agree the answer is no,
> think about what that implies for the relationship between
> probability and fuzzy.
>
> : Kindest regards,
> : Barry.
>
> Regards,
> S. F. Thomas
>
>