**Subject: **BISC: A Challenge to Bayesians (augmented version)

**From: **Michelle T. Lin (*michlin@eecs.berkeley.edu*)

**Date: **Thu Aug 03 2000 - 16:28:15 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 (augmented version)

-------------------------

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.

What was said above may be viewed as an argument suggesting

that probability theory is in need of upgrading through

generalization. But what has to be recognized, in addition, is that

upgrading of probability theory is necessary but not sufficient.

Specifically, there is a widely held belief that probability theory,

upgraded or not, is sufficient for dealing with any or all issues

which relate to partial certainty or incompleteness of information.

But what is becoming increasingly clear is that this is not the case.

The growing complexity of problems which arise in the conception,

design and utilization of information/intelligent systems requires

that all relevant methodologies be marshalled for solution. What this

points to is a need for formation of a coalition of methodologies

which, in combination, are much more powerful than they are in a

stand-alone mode. Such a coalition or consortium is soft computing --

a synergistic collection of computationally-oriented methodologies

whose principal members are fuzzy logic, neurocomputing, evolutionary

computing, probabilistic computing, chaotic computing and machine

learning theory. The obvious superiority of soft computing over any

one of its constituents suggests that, in the future, most

information/intelligent systems will be of hybrid type, employing

various combination of methodologies to deal with uncertainty,

possibility, imprecision, incompleteness, partial understanding and

partial truth.

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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**Next message:**Ning Zhong: "Web Intelligence (WI'2001): Call for Papers"**Previous message:**Ning Zhong: "Web Intelligence (WI'2001): Call for Papers"**Next in thread:**MVCS: "Re: BISC: A Challenge to Bayesians (augmented version)"**Reply:**MVCS: "Re: BISC: A Challenge to Bayesians (augmented version)"**Reply:**Michiel de Roo: "Re: BISC: A Challenge to Bayesians (augmented version)"

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