FUZZY SET THEORY vsSUBJECTIVE PROBABILITY
1 INTRODUCTION
Around the middle of seventeenth century, the formal concept of numerical
probability was established. Since that time, and till mid-twentieth
century, uncertainty faced an absolute monopoly from probability theory
which was viewed by several techniques. The robability life is long
enough (365 years old) to have establish its syntactic, semantic, and
pragmatic foundation. Due to the long probability monopoly, a seemingly
obvious connection with uncertainty has never been questioned. During the
1950's, an alternative theories began to emerge. apers continued
announcing new concepts till the 1970s. Of course a series of papers
announcing deferent concepts in dealing with uncertainty such as those in
Table 3.1, expresses a total disapproval for the monopoly of the
probability theory dealing with uncertainty.
Chquet Capacities. 1954
Fuzzy Sets. 1987-199
Fuzzy Measures. 1992
Random Sets. 1975
Rough Sets. 1991
Imprecise Probability. 1987199
Evidence Theory. 1976
Fuzzy Rough Sets. 1990
Modal Logic Conceptualized uncertainty. 1992
Principle of Uncertainty invariance. 1993-1990-1992
Table 3.1 Quick review of concepts dealing with uncertainty.
In response to this revolution, series of papers were published in the
1980's defending probability
theory as the only correct formal model for representing uncertainty, and
reject all other
alternative theories for modeling uncertainty. A (for-against) approach
has been used to assert,
stand in a position, or debate new concepts and theories dealing with
uncertainty, specially fuzzy
set theory. This approach resulted in the appearance of series of papers
that helped man to have
better understanding of uncertainty and its applications.Perhaps the
disadvantage of this series
is that they are rarely found to be consistent, there are very few terms
that all writers agree on.
For terms that seem logical and quite obvious, there are many researchers
who are skeptical
about, and they usually have respectable reasons behind their opinion.
Another draw back of this
series, is that papers are usually one to one controversy debates, which
need intensive
gathering, and organizing. The main skeleton of such studies usually, to
some extent, is to take
one of two positions;(1)Non-Statistical uncertainty does not exist, and
so, probability is the best
theory to model the existing statistical uncertainty. While fuzzy set
theory can only contribute as
one clever face of probability.(2)Non-Statistical uncertainty exists, but
probability can still share
fuzzy set theory in modeling it. Moreover, it can do as good as fuzzy set
theory.
Assuming that "Non-statistical uncertainty really does not exist".
Certainly there is a reluctance
on the part of mathematicians, and statisticians, in particular, to agree
that there is more than
one kind of uncertainty. Also it is expected that fuzzy opponents provide
one view of probability,
while in fact there are more than one view of probability itself.First,
suppose that non-statistical
uncertainty does not exist, is there in probability a system that
everyone agrees on,
mathematically or philosophically?. Not really, still there are nice
variety of choices. For
example, the view and use of probability can be based on the relative
frequency concept; or the
subjectivist idea, or the axiomatic approach [A0X11]. This argument
within probability is hardly
resolved.Now, if it is intended to abandon fuzzy models on the premise
that all uncertainty is
essentially statistical. Is this enough solid ground to drive on?.
Apparently not, for probabilists
themselves continue to debate the philosophical and mathematical under
pinning of there
subject, approximately 365 years after Galileo first wrote about laws of
chances in connection
with gambling. Now which of these deferent perspectives about probability
should be taken, and
how could it guide to solve problems exhibiting uncertainty?. The choice
here, within probability,
is not a sharp cut and dried as probabilists suggest.Since it is now
clear that deferent kinds of
uncertainty exist, we can expect to find deferent model for each.
Following Zadeh's (1965) idea,
it is obvious that non-statistical uncertainty exists, and that it is
due, and among other causes
perhaps, to linguistic imprecision.Among various disciplines,
mathematicians, and in particular
statisticians, are the most fervent protagonists in the resistance of
fuzzy set theory. Thus, the
basic premise for this chapter is to answer a set of questions;
1-What is the set of hypotheses that distinguish fuzzy set theory from
probability theory?
2-If it is agreed that real world is vague and imprecise, if so, should
it be modeled by prescribing
using a precise mathematical model that forces man to remove ambiguity,
or by emulating
human thinking?
3-If subjective problems, such as prediction problems, are probability,
or fuzzy oriented?
4-If natural language is a convenient vehicle to drive man believes into
mathematical model?
5-If both probability, and fuzzy set theory are human constructs, or
fuzzy set theory should be
distinguished from probability theory on the ground that it is a real
world theory like theories of
physics and astronomy?
6-If there exist an axiomatic proof for each approach?
7-If fuzzy set theory is only a view of probability theory in a clever
way, or it is a super theory that
includes probability underneath?
8-If there is a protocol or an operational paradigm for the fuzzy set
theory that can be applied
independently.
The previous set of questions are discussed in this here. Question 2
through 8 are discussed in
section 3.2 through 3.8 respectively. Perhaps the titles of sections 2
through 8 represent an
answer to question number one. Then, a short analysis for types and
sources of vagueness is
included in section 3.9. The chapter ends with section 3.10 which
includes an evaluation, of the
contradicting opinions.
2 THE REALITY HYPOTHE
SIS:The need for representing and modeling uncertainty comes from the
fact that, for any
theory, the more it represents the real world, the better the results it
provides. Representing
uncertainty is rather arguable. Many probability advocates claim that
nothing is uncertain in the
real world.The argument of reality hypothesis was raised because of the
vagueness found in the
real world. The argument, in a side suggest that, the real world is
precise and definite, while
uncertainty is in man's perception to the real world facts. Thus, "one
can not appropriately fit the
real world into a classical mathematical model [A1X17]ö. On the other
side, fuzzy theorists think
that "most phenomena are inherently vague, and therefore, do not lend
themselves to analysis
by systems requiring precise definition [A1X0]ö, which means that, it can
be true if "imprecision
is an inherent property of the world external to an observer [A1X0]ö. As
a result of the above,
this section holds a discussion; if the imprecision resides in the real
world; or in the
observer.Many examples were placed to discuss this argument. Zadeh
(1965) [A1X02] gave an
example of organism having ambiguous status with respect to the class of
animal. Other popular
examples like "how long is a long street? [A1X13]ö, and what is the speed
of a fast car, will not
be discussed here. A picture of an imperfect ellipse (actually a hand
drown oval) was initially
raised by Kosko (1990) [A1X15], then he asked a question, "does it make
more sense to say that
the oval is probably a circle or ellipse, or that it is a fuzzy ellipse?"
[A1X15].If, for example, it is
required to classify objects unambiguously, such as distinguishing an
ellipse from ovals, Two
approaches may be proposed; (1) having one classification for all cases,
or (2) having one
classification for each case. Since both approaches are uninformative,
compromise could be
established by using minimal number of categories, each of which is reach
enough to enable a
distinguish among classes to the extent required on practice. To this
point the overlap between
proposed classification can be handled by probability [A1X0] [A1X14].
Each of the above
techniques can be considered as a statistical technique for solving the
problem of classifying
(distinguishing) ambiguous classes, as proposed by [A1X0][A1X14].Applying
the above statistical
technique to distinguish between an oval and an ellipse is far away from
the problem. "There is
nothing random about the matter. The situation is deterministic, all
facts are in, but yet,
uncertainty remains. The uncertainty is due to the occurrence of two
properties; (1) to some
extent, the inexact oval is an ellipse, and (2) to some extent, it is not
an ellipse [A1X15]ö.The
previous explanation is still arguable. The uncertainty exists but is it
in the object itself (the oval),
or in the eye of the beholder? The claim is still there. Uncertainty of
objects like beauty, exists
but in the eye of the beholder. The subjective probability measures such
uncertainty according to
well established standards such as this based on urn models; or
indifference between gambles
[A1X0]. Furthermore, fuzzy opponents claim that the previous statistical
technique could solve
the problem of the ellipse by assigning higher probability of the
figure's being an ellipse than
being any thing else. As a result of this the question can be answered by
saying that, the figure is
probably an ellipse, where probability here refers to the observer's
personal uncertainty [A1X0].At
this point, a reference to principles would clarify the misleading
claims. The probability that an
imperfect ellipse is an ideal ellipse is zero. While using a degree of
membership for an imperfect
ellipse would give a very high membership value (almost one) [A2X0].The
assumption that
considers all of real world classifications should be crisp, lead to a
situation of almost empty
classes. In a sense, the problem of an imperfect ellipse could be divided
into two deferent steps:
(1)assessing the proximity of a figure to an ideal ellipse, and
(2)decision problem. The proximity
problem can be solved using any mathematical function to compare it with
the closest ellipse.
Thus, the proximity problem involves no decision step, and fuzzy
classification is clearly
appropriate, while probability can not be involved. The decision problem
might involve
probability. For instance, the probability that an individual will act as
if it were an ellipse.
[A2X0]While it seems reasonable to postulate that "the closer the
figure to an ellipse, the higher
probability that an individual will act as if the figure were an
ellipse". But, this probability should
not necessarily be equated to the degree of proximity of the figure to
ideal ellipse [A2X0].In
other sense, the example initially raised by Kosko (1990) [A6X01] aims to
discuss the vague
definition of an ellipse. Thus, there is no point here of referring an
ellipse to the ideal perfect
ellipse (geometrically). There is no doubt that discussing the fact that
a given ellipse has non-
zero width, of the drown curve, is out of the point.Subjective
probability offered a statistical
technique for solving classification problems, such as the problem of
imperfect ellipse, which
suggests that Bayesian probability of being an ellipse is non-zero. So,
the technique presumably
can not be using a strict mathematical definition of an ellipse,
otherwise the probability should
have been zero. Thus, what does probability mean by the word ellipse
then. It is meaningless to
talk to someone's Bayesian probability of figure being an ellipse, since
the latter proposition is
undefined. [A6X0]The claim that imprecision is not an inherent property
in the world external to
observer, leads to ask, what is meant by "world external to observer"?.
Is vagueness of language
part of the world external to observer?. Vagueness of language is a
property of our world; in that
sense, it is a property of the world we live in, since language is
fundamental to our perception of
reality, and so is very much part of our world. Vagueness of a concept
does not completely
reside in the observer; the point of language is that it is shared; when
a vague word like "tall" or
"ellipse" is used, given that context is very clear, you even still have
vague meaning. [A6X0 sec
III]The real world has more ambiguous objects than precise. Ambiguity and
vagueness resides in
the real world more than the observer. Vagueness came out of many
sources, language
ambiguity is the most common, other sources are discussed later in this
chapter. Observer
usually try to catch facts of the real world using tools available to
him, but due to vagueness of
world and his tools, he fail to catch world reality. Instead he catch
uncertain facts about the
world, so he needs to analyze with systems accepting indefinite
definitions.Moreover,
transferring the world into precise crisp sets leads to almost empty
sets. In other words, the
phrase "since no yard-stick is exactly one yard long, yard-sticks are
useless !! [A1X17]", is quite
absurd.