Probability Theory Needs an Infusion of Fuzzy Logic to Enhance its
Ability to Deal with Real World Problems
Discussions and debates centering on the relationship between
fuzzy set theory and fuzzy logic, on one side, and probability theory,
on the other, have a long history starting shortly after the
publication of my first paper on fuzzy sets (1965). There is an
extensive literature on this subject, including several papers of
mine, of which the latest [Zadeh 1995] appears in a special issue of
Technometrics that focuses on the relationship in question.
The following is an updated expression of my views. Your
comments would be appreciated. Please do not hesitate to disagree
with me.
First, a point of clarification. By probability theory in the
heading of this message is meant standard probability theory (PT) of
the kind found in textbooks and taught in courses. By infusion is
meant generalization. Thus, the crux of my argument is that PT is in
need of generalization, first by f-generalization (fuzzification) and
second by g-generalization (granulation). In combination,
f-generalization and g-generalization give rise to f.g-generalization
(fuzzy granulation). F-generalization of PT leads to what might be
denoted as PT+. Then, g-generalization of PT+ leads to PT++, which
may be viewed as f.g-generalization of PT. Basically, fuzzy
granulation reflects the finite cognitive ability of humans to resolve
details and store information. In summary, my contention is that to
enhance its effectiveness PT should be generalized to PT++. This has
been done already to a significant extent by a number of contributors,
but much more remains to be done.
What is not in dispute is that standard probability theory
(PT) provides a vast array of concepts and techniques which are
effective in dealing with a wide variety of problems in which the
available information is lacking in certainty. But what may come as a
surprise to some is that along with such problems stand many very
simple problems for which PT offers no solutions. Here is an sample.
It should be noted that the underlying difficulties in these problems
are well-known to probability theory professionals.
DB 1. What is the probability that my tax return will be
audited?
DB 2. What is the probability that my car may be stolen?
DB 3. How long does it take to get from the hotel to the
airport by taxi?
DB 4. What is the probability that Mary is telling the truth?
DB 5. A and B played 10 times. A won 7 times. The last 3 times
B won. What is the probability that A will win?
Questions of this kind are routinely faced and answered by
humans. The answers, however, are not numbers; they are linguistic
descriptions of fuzzy perceptions of probabilities, e.g., not very
high, quite unlikely, about 0.8, etc. Such answers cannot be arrived
at through the use of PT. What is needed for this purpose is PT++.
I discussed Problems 1-5 with Professor David Blackwell (UC
Berkeley), who is one of world's leading authorities on probability
theory. The initials DB signify that he agrees with my assessments.
What are the sources of difficulty in using PT? In Problem 1,
the difficulty comes from the basic property of conditional
probabilities, namely, given p(X), all that can be said is that the
value of p(X|Y) is between 0 and 1. Thus, if I start with the
knowledge that 1% of tax returns are audited, it tells me nothing
about the probability that my tax return will be audited. The same
holds true when I add more detailed information about myself, e.g., my
profession, income, age, place of residence, etc.. IRS may be able to
tell me what fraction of returns in a particular category are audited,
but all that can be said about the probability that my return will be
audited that is between 0 and 1.
As I have alluded to earlier, when I prepare my tax return, I
do have a fuzzy perception of the chance that I may be audited. To
arrive at this fuzzy perception what is needed is fuzzy logic, or,
more specifically, PT++. I will discuss in a later installment how
this can be done through the use of PT++.
In Problems 3 and 5, the difficulty is that we are dealing
with a time series drawn from a nonstationary process. In such cases,
probabilities do not exist. Unfortunately, this is frequently the
case when PT is used.
In Problem 3, when I pose the question to a hotel clerk, he
may tell me that it would take approximately 20-25 minutes. The point
is that the answer could not be deduced in a rigorous way by applying
PT. Again, to arrive at clerk's answer what is needed is PT++.
In Problem 4, the difficulty is that truth is a matter of
degree. For example, if I ask Mary how old she is, and she tells me
that she is 30 but in fact is 31, the degree of truth might be 0.9.
On the other hand, if she tells me that she is 25, the degree of truth
might be 0.5. The point is that the event "telling the truth" is, in
general, a fuzzy event, PT does not support fuzzy events.
Another class of simple problems which PT cannot handle relate
to common sense reasoning exemplified by:
6. Most young men are healthy; Robert is young.
What can be said about Robert's health?
7. Most young men are healthy; it is likely that Robert is young.
What can be said about Robert's health?
8. Slimness is attractive; Cindy is slim.
What can be said about Cindy's attractiveness?
In what ways do PT+ and PT++ enhance the ability of
probability theory to deal with real world problems? In relation to
PT, PT+ has the capability to deal with:
1. fuzzy events, e.g., warm day
2. fuzzy numbers, quantifiers and probabilities, e.g., about 0.7,
most, not very likely
3. fuzzy relations, e.g., much larger than
4. fuzzy truths and fuzzy possibilities, e.g., very true and
quite possible
In addition, PT+ has the potential -- as yet largely unrealized -- to
fuzzify such basic concepts as independence, stationarity and
normality.
PT++ adds to PT+ further capabilities which derive from the
use of granulation. They are, mainly:
1. linguistic (granular) variables
2. fuzzy rule sets and fuzzy graphs
3. granular goals and constraints
At this juncture there exists an extensive and growing
literature centering on the calculi of imprecise and fuzzy
probabilities. The concepts of PT+ and PT++ serve to clarify some of
the basic issues in these calculi and suggest directions in which the
ability of probability theory to deal with real worlds problems may be
enhanced.
Reference:
L. A. Zadeh, "Probability Theory and Fuzzy Logic are Complementary Rather
Than Competitive," Technometrics, vol. 37, pp. 271-276, 1995.
Warm regards to all,
Lotfi
P.S. Please confirm receipt via e-mail to me. Please include your
address, tel,
fax, etc.
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