BISC: A Challenge to Bayesians


Subject: BISC: A Challenge to Bayesians
From: Michelle T. Lin (michlin@eecs.berkeley.edu)
Date: Sun Jul 30 2000 - 21:57:20 MET DST


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Berkeley Initiative in Soft Computing (BISC)
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To: BISC Group
From: L. A. Zadeh <zadeh@cs.berkeley.edu>

A Challenge to Bayesians
-------------------------

        The past two decades have witnessed a dramatic growth in the
use of probability-based methods in a wide variety of applications
centering on automation of decision-making in an environment of
uncertainty and incompleteness of information.
        
        Successes of probability theory have high visibility. But what
is not widely recognized is that successes of probability theory mask
a fundamental limitation -- the inability to operate on what may be
called perception-based information. Such information is exemplified
by the following. Assume that I look at a box containing balls of
various sizes and form the perceptions: (a) there are about twenty
balls; (b) most are large; and (c) a few are small. The question is:
What is the probability that a ball drawn at random is neither
large nor small? Probability theory cannot answer this question
because there is no mechanism within the theory to represent the
meaning of perceptions in a form that lends itself to computation. The
same problem arises in the examples:

--Usually Robert returns from work at about 6 pm. What is the
  probability that Robert is home at 6:30 pm? What is the earliest
  time at which the probability that Robert is home is high?

--I do not know Michelle's age but my perceptions are: (a) it is very
  unlikely that Michelle is old; and (b) it is likely that Michelle is
  not young. What is the probability that Michelle is neither young
  nor old?

--X is a normally distributed random variable with small mean and
  small variance. What is the probability that X is neither small nor
  large?

--X and Y are real-valued variables, with Y=f(X). My perception of f
  is described by (a) if X is small then Y is small; (b) if X is
  medium then Y is large; (c) if X is large then Y is small. X is a
  normally distributed random variable with small mean and small
  variance. What is the probability that Y is much larger than X?

--X and Y are random variables taking values in the set
  U={0,1,...,20}, with Y=f(X). My perception of the probability
  distribution of X, p, is described by: (a) if X is small then
  probability is low; (b) if X is medium then probability is high; (c)
  if X is large then probability is low. My perception of f is
  described by: (a) if X is small then Y is large; (b) if X is medium
  then Y is small; (c) if X is large then Y is large. What is the
  probability distribution of Y? What is the probability that Y is
  medium?

--Given the data in insurance company database, what is the
  probability that my car may be stolen? In this case, the answer
  depends on perception-based information which is not in insurance
  company database.

--I am staying at a hotel and have a rental car. I ask the concierge
  "How long would it take me to drive to the airport?" Concierge
  answers "About 20-25 minutes." Probability theory cannot answer the
  question because the answer is based on perception-based
  information.
  
        In these simple examples -- examples drawn mostly from
everyday experiences -- the general problem is that of estimation of
probabilities of imprecisely defined events, given a mixture of
measurement-based and perception-based information. The crux of the
difficulty is that perception-based information is usually described
in a natural language -- a language which probability theory cannot
understand and hence is not equipped to handle.

        My examples are intended to challenge the unquestioned belief
within the Bayesian community that probability theory can handle any
kind of information, including information which is perception-based.
However, it is possible -- as sketched in the following -- to
generalize standard probability theory, PT, in a way that adds to PT a
capability to operate on perception-based information. The
generalization in question involves three stages labeled: (a)
f-generalization; (b) f.g-generalization: and (c)
nl-generalization. More specifically:

        (a) f-generalization involves fuzzification, that is,
progression from crisp sets to fuzzy sets, leading to a generalization
of PT which is denoted as PT+. In PT+, probabilities, functions,
relations, measures and everything else are allowed to have fuzzy
denotations, that is, be a matter of degree. In particular,
probabilities described as low, high, not very high, etc. are
interpreted as labels of fuzzy subsets of the unit interval or,
equivalently, as possibility distributions of their numerical values.

        (b) f.g-generalization involves fuzzy granulation of
variables, functions, relations, etc., leading to a generalization of
PT which is denoted as PT++. By fuzzy granulation of a variable, X,
what is meant is a partition of the range of X into fuzzy granules,
with a granule being a clump of values of X which are drawn together
by indistinguishability, similarity, proximity, or functionality. For
example, fuzzy granulation of the variable Age partitions its
values into fuzzy granules labeled very young, young, middle-aged,
old, very old, etc. Membership functions of such granules are usually
assumed to be triangular or trapezoidal. Basically, granulation
reflects the bounded ability of the human mind to resolve detail and
store information.

        (c) nl-generalization involves an addition to PT++ of a
capability to represent the meaning of propositions expressed in a
natural language, with the understanding that such propositions serve
as descriptors of perceptions. nl-generalization of PT leads to
perception-based probability theory denoted as PTp.

        Perception-based theory of probabilistic reasoning suggests
new problems and new directions in the development of probability
theory. It is inevitable that in coming years there will be a
progression from PT to PTp; since PTp enhances the ability of
probnability theory to deal with realistic problems in which
decision-relevant information is a mixture of measurements and
perceptions.
        
        In summary, contrary to the central tenet of Bayesian belief,
PT is not sufficient for dealing with realistic problems. What is
needed for this purpose is PTp.
        
        
                                Warm regards to all,
                                
                                Lotfi
                                
----------------------------------------------------------
Lotfi A. Zadeh
Professor in the Graduate School and Director,
Berkeley Initiative in Soft Computing (BISC)
CS Division, Department of EECS
University of California
Berkeley, CA 94720-1776
Tel/office: (510) 642-4959 Fax/office: (510) 642-1712
Tel/home: (510) 526-2569 Fax/home: (510) 526-2433
email: zadeh@cs.berkeley.edu
http://www.cs.berkeley.edu/People/Faculty/Homepages/zadeh.html
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To post your comments to the BISC Group, please send them to
me(zadeh@cs.berkeley.edu) with cc to Michael Berthold
(berthold@cs.berkeley.edu)
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