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From: sthomas@decan.gate.net (S. F. Thomas)
Subject: Re: Fuzzy theory or probability theory?
Message-ID: <1994Dec4.010347.23979@decan.gate.net>
Organization: Decision Analytics, Inc.
Date: Sun, 4 Dec 1994 01:03:47 GMT
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caj@jerry.psu.edu wrote:
(( cuts ))
: I personally have never attempted to estimate a chance probability using
               ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
: fuzzy logig theory.  Those that do, do not have my support.
  ^^^^^^^^^^^^^^^^^^

Maybe not "fuzzy logic theory", but if you use language, you use 
terms commonly regarded as fuzzy -- such as "high", "low", etc. --
as no less a luminary than Sir John Hicks has done, while arguing
for the inapplicability, in some situations, of the frequentist 
concept of probability.  See the following excerpt from
"Fuzziness and Probability" (from Chap. II -- "Phenomena, Models
and Probability"):


   "Prevailing views of probability may be divided broadly
   into the frequency view, and the
   axiomatic view (Kendall, 1947, p.165; Hicks, 1979, p.105). It is
   conventional to adopt the criterion of repeatability as dividing
   the respective areas of application of the two views. On the
   frequency view, the paradigm is one of trials or experiments that
   can be repeated, allowing the probability of an event to be
   identified with the relative frequency with which it occurs on
   repeated trials. Differences in the outcome from trial to trial
   are ascribed to random chance, as in drawing cards from a deck,
   tossing dice, observing crop yield in experimental plots, quality
   testing of manufactures, and so on. The axiomatic view is a
   response to situations where an uncertain event or proposition is
   contemplated, but such an event is considered to be part of a
   unique trial, inherently nonrepeatable. Hicks (1979, p.107) gives
   an example attributed to Cramer: the probability, considered in
   1944, that the European war would come to an end within a year.
   This, it is maintained, is not a matter of trials that can be
   repeated. Hicks goes on to say that economic behavior, as well as
   economic theory, deals with probabilities which cannot be
   interpreted in terms of random experiments. Hicks, the economist,
   is therefore well disposed towards a broader, axiomatic concept of
   probability as suggested by Jeffreys (1939, p.15): "... a valid
   primitive idea expressing the degree of confidence that we may
   reasonably have in a proposition." Taking this notion of
   degree of confidence, or belief, as primitive, the axiomatic view
   proceeds to postulate a series of axioms for this primitive
   concept, which allows the derivation of a calculus that obeys all
   the laws of probability familiar from the frequency view.
   Implicit in the decision to take probability as a primitive
   concept, is the judgment that it is impossible or unprofitable to
   take any more primitive concept as fundamental. The frequency
   view, as we have seen, does take the notion of repeated trials as
   more fundamental, but this seemingly limits the field of
   application of the probability concept to areas where we can
   convince ourselves that the trials in question may indeed be
   repeated.
   
   I believe it is profitable to extend the 
   frequency concept even to those areas of application where
   it must be admitted that the obvious 
   trial in question could not possibly be repeated. 
   Obviously, we could never go backwards in history 
   to repeat something like the European war from 
   1944 onwards to see what fraction of the time it 
   would come to an end within one year of some 
   appropriate date in 1944. However, I believe it is 
   fruitful to reflect that there is indeed some 
   broader phenomenon at issue, of which the European 
   war is but one instance. This broader phenomenon 
   is the phenomenon of war, of which there are 
   countless examples in history providing a basis 
   for generalization. The European war could 
   therefore be regarded as one individual instance 
   within a larger population of instances of the 
   broader phenomenon of war. 
   
   More generally, I would claim that the criterion
   of repeatability as a way of dividing the 
   respective areas of application of the frequency 
   view and the axiomatic view does not have the 
   validity that it would seem to have on the 
   surface. In the first place, no experiment may 
   literally be repeated, since things change
   (time, at a minimum) from one experiment to the next. If 
   we make another throw of a pair of dice, or pick another card
   from a pack, or grow another crop on an experimental plot, we
   have not repeated an earlier trial; at the very least, time has
   intervened, and at worst, some determinant of the response
   variable may have changed, accounting for the changed result. Yet
   it is meaningful to talk of a repetition, for there is something
   that stays the same from one trial to the next. That something is
   what I would call the "morphology" we mentally construct around the
   phenomenon. In typical frequentist statistical experiments, such
   a morphology is quite explicit -- the population of occurrences
   from which observations are drawn is well defined, and the
   variables of interest, response variables and explanatory
   variables are well defined. In the case of the question
   concerning the European war, nothing is defined -- neither the
   population of occurrences of which the European war may be deemed
   to be an instance, nor the precise variables that we should look
   at to help us make a judgment and answer the question posed.
   Nevertheless, when we engage in discourse concerning the
   phenomenon, we implicitly construct such a morphology. 
   One may offer some such discussion as follows: in any war, it is the
   relatively stronger of the combatants that usually wins, and the
   greater the relative strength of one combatant over the other,
   the faster that combatant wins. The strength of a combatant
   depends upon the number and quality of men, weapons, and war
   materiel that it possesses, and the cohesiveness and morale of
   the troops, as well as on the strategic deployment of such forces.
   And so on. It is not my intention here to develop a theory of
   war or combat, only to illustrate my point that in the
   elaboration of discourse concerning a phenomenon, we implicitly
   construct a mental morphology around it, defining a population of
   "objects" -- combatants, considered in pairs -- and a set of
   attributes or variables which enter either as response variables
   of interest -- relative strength in this case -- or as explanatory
   variables -- number and quality of men, etc. Once we have such a
   morphology, we have the basis for a frequentist approach to
   probabilistic modeling of the phenomenon, for it allows us to
   speak in general terms about events that would otherwise be
   considered simply unique. Thus, on the one hand, the
   repeatability of unarguably repeatable statistical experiments
   really derive from the morphology of the situation, and on the
   other hand, once we do construct a morphology for seemingly
   unique events, they also become individual instances, or
   repetitions, of occurrences that fit into a larger pattern. I
   conclude that morphology is the key concept that allows us to
   close the gap between the frequency view and the axiomatic view
   in the application of the probability concept.

         ...

   What I am saying, in other words, is
   that the criterion for the definition of probability is not
   repeatability -- one cannot step into the same river twice ...
   at the very least time and molecules would have changed -- but
   the existence of an assumed morphology, perhaps implicit, with
   reference to which particular instances, all distinct, may be
   observed. What repeats is the morphology, not the particular
   instances within the morphology.
   
   It should be apparent from the foregoing that the concept of
   probability is distinct from its measurement: it is one thing to
   suggest a morphology for some phenomenon -- such as the duration
   of wars; it is another to estimate the probability of some event
   over the extension set of instances of that phenomenon, within
   the assumed morphology. Probability measurement is subject to the
   same fuzziness in general as measurement of any other attribute.
   In Hicks' discussion, the burden of his argument is to agree with
   other writers (Keynes, Jeffreys) that the frequency theory of
   probability is not wide enough for economics, yet to disagree
   with the ordering axiom in Jeffrey's development of a subjective
   probability theory. Hicks summarizes his position in the
   following way:
   
      We may take as the general
      definitions of probability Jeffrey's 'primitive notion' --
      the degree of confidence that may reasonably
      be expressed in a proposition on the basis of given evidence. But
      we must then rewrite his first axiom. We must say that of two
      alternatives, on given evidence, either A is more probable than
      B, or B is more probable than A, or they are equally probable, or
      that they are not comparable. That makes probabilities partially,
      but not completely, orderable.
      (Thinking of the analogy with the indifference curves
      of utility theory, our indifference curve may be thick).
      --- John Hicks, _Causality_in_Economics_, 1975   
   
   I agree with Hicks on one point -- that the 
   total ordering axiom does not, as an empirical 
   matter, always, or even usually, apply. Measurement is
   fuzzy in general. However, remarks I have 
   made earlier (Section 2) should make it clear that 
   I think it is important to distinguish the
   abstract properties of the attribute space, where, 
   for the attribute of probability, total ordering 
   does apply, from the empirical idiosyncracies of 
   the judge or measurement device used for carrying 
   out measurements according to the attribute. It is 
   in this empirical/perceptional domain that comparability problems arise.
   As we have seen, the 
   notions of fuzzy set and fuzzy measurement provide 
   a way of mediating between the two.
   
   Where I do not agree with Hicks is in adopting Jeffrey's
   definition of probability as the degree of confidence that may
   reasonably be expressed in a proposition on the basis of given
   evidence. I prefer the notion of probability developed so far in
   this essay -- morphological probability, if you will -- because to
   my mind it captures both the frequentist concept so useful in the
   physical sciences, and the subjective concept put forward by
   Jeffreys. In estimating a morphological probability, the
   distinction between objective (frequency) and subjective
   probability does not apply. If one is dealing with a finite
   extension set, and one has counted and classified all elements of
   the extension set according to attribute value, one would have an
   exact probability or statistical model for the extension set. If,
   however, one is dealing either with an infinite extension
   set, or with limited sample information from a finite extension
   set, any inferences or assertions made regarding 'the'
   probability of any event within the sample space will be mere
   hypothesis, and subject to uncertainty. Whether such hypothesis
   is based on a subjective, 'degree of confidence' attitude, or on
   some kind of 'objective' goodness-of-fit procedure based on an
   observed sample is irrelevant to the concept of morphological
   probability. In either case, a postulated probability model would
   be in general purely a hypothesis one is prepared to entertain as
   having some descriptive accuracy in summarizing the distribution
   of occurrences over an extension set. Furthermore, when a
   subjective probability hypothesis is stated in terms that reflect
   the uncertainty that one feels (e.g. 'most rich people are happy')
   we are presented with a fuzzy range of (exact)
   probability models which may apply. Likewise, when sample data
   are available, the likelihood function in effect also defines a
   fuzzy range of probability models consistent with the sample
   data. Thus the distinction often made between subjective
   probability and frequency probability loses much of its force
   when the construct of morphological probability is applied."


(( cuts ))

: I offer that I will place objects that are all red and circular in Set A.

: If have an object that is red but is oval conventional set theory does
: not allow me to put it in set A.  If I did I would not be able to distinguish
: it from actual circular objects.  But a generalization allows me to say that
: it is very similar to objects in set A.  How do I describe this generalization?
: Certainly this is not probability problem and I am not looking to predict
: anything.  I am simply wondering how to put this odd object into one of
: my specifically designed sets.

: I use fuzzy set theory to give this object a less than 1 membership in set A.
: It is a simple and straightforward solution that is very easy to understand
: and use.  I may validate my using fuzzy set theory here simply for such
: practical reasons but I also happen to believe this type of solution is a
: very common human method.

: Why would probability theory do better here since this is not a chance
: probability problem?  I am not concerned with the chances of finding a red
: and circular object.  I am concerned with how to classify this object that
: does not exactly fit my set description.  
(( cuts ))
: How would you classify this object while recognizing that it is very similar
: to the ideal member for this set?

This is a job for language.  If you subscribe to the conventions of
language, you presumably want to assign a degree of membership which
bears some relationship to language use ... you want the probability
that some randomly chosen competent speaker of the language would assign
your red oval to the (fuzzy) set of red, circular objects, in some
context that is appropriate.  Now, if you are _not_ concerned about
language use in the population at large, and how the population at
large would classify your object, you can in effect define your own
"language", and you are of course then quite free to assign any membership
value you choose, using any criterion or metric you care to.  In either
case, the broad semantics of fuzzy set theory seem very much applicable
to me, notwithstanding the identification of the membership value 
with a probability value in the first case.  In that case, a series 
of such probability values defined over a series of candidate "red,
circular" objects defines, not a probability distribution, but a 
likelihood, or, membership function.

: Clark A. Janes

Just my opinion.

Cheers!
S.F.Thomas
