# BISC: Zadeh/ Seminar on Mathematical Economics

From: Masoud Nikravesh (nikravesh@eecs.berkeley.edu)
Date: Thu Mar 15 2001 - 10:47:01 MET

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Berkeley Initiative in Soft Computing (BISC)
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Seminar on Mathematical Economics
Thursday, March 15, 2001
Room 639 Evans
4:00-5:00pm

Toward a Perception-Based Theory of Probabilistic Reasoning
Abstract

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 6:00 p.m. What is the probability that
Robert is home at 6:30 p.m.?
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 large?
Given the data in an insurance company database, what is the probability that my
car may be stolen? In this case, the answer depends on perception-based
information that is not in an insurance company database.
In these simple examples--examples drawn 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 that probability theory cannot understand and hence
is not equipped to handle.
To endow probability theory with a capability to operate on perception-based
information, it is necessary to generalize it in three ways. To this end, let PT
denote standard probability theory of the kind taught in university-level
courses. The three modes of generalization are 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 that 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 that 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 that are drawn together by
indistinguishability, similarity, proximity, or functionality. For example,
fuzzy granulation of the variable age partitions its vales 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; and (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.
An assumption that plays a key role in PTp is that the meaning of a proposition,
p, drawn from a natural language may be represented as what is called a
generalized constraint on a variable. More specifically, a generalized
constraint is represented as X isr R, where X is the constrained variable; R is
the constraining relation; and isr, pronounced ezar, is a copula in which r is
an indexing variable whose value defines the way in which R constrains X. The
principal types of constraints are: equality constraint, in which case isr is
abbreviated to =; possibilistic constraint, with r abbreviated to blank;
veristic constraint, with r = v; probabilistic constraint, in which case r = p,
X is a random variable and R is its probability distribution; random-set
constraint, r = rs, in which case X is set-valued random variable and R is its
probability distribution; fuzzy-graph constraint, r = fg, in which case X is a
function or a relation and R is its fuzzy graph; and usuality constraint, r = u,
in which case X is a random variable and R is its usual--rather than
expected--value.
The principal constraints are allowed to be modified, qualified, and combined,
leading to composite generalized constraints. An example is: usually (X is
small) and (X is large) is unlikely. Another example is: if (X is very small)
then (Y is not very large) or if (X is large) then (Y is small).
The collection of composite generalized constraints forms what is referred to as
the Generalized Constraint Language (GCL). Thus, in PTp, the Generalized
Constraint Language serves to represent the meaning of perception-based
information. Translation of descriptors of perceptions into GCL is accomplished
through the use of what is called the constraint-centered semantics of natural
languages (CSNL). Translating descriptors of perceptions into GCL is the first
stage of perception-based probabilistic reasoning.
The second stage involves goal-directed propagation of generalized constraints
from premises to conclusions. The rules governing generalized constraint
propagation coincide with the rules of inference in fuzzy logic. The principal
rule of inference is the generalized extension principle. In general, use of
this principle reduces computation of desired probabilities to the solution of
constrained problems in variational calculus or mathematical programming.
It should be noted that constraint-centered semantics of natural languages
serves to translate propositions expressed in a natural language into GCL. What
may be called the constraint-centered semantics of GCL, written as CSGCL, serves
to represent the meaning of a composite constraint in GCL as a singular
constraint X isr R. The reduction of a composite constraint to a singular
constraint is accomplished through the use of rules that govern generalized
constraint propagation.
Another point of importance is that the Generalized Constraint Language is
maximally expressive, since it incorporates all conceivable constraints. A
proposition in a natural language, NL, which is translatable into GCL, is said
to be admissible. The richness of GCL justifies the default assumption that any
given proposition in NL is admissible. The subset of admissible propositions in
NL constitutes what is referred to as a precisiated natural language, PNL. The
concept of PNL opens the door to a significant enlargement of the role of
natural languages in information processing, decision, and control.
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 probability theory to deal with realistic problems in which
decision-relevant information is a mixture of measurements and perceptions.
Lotfi A. Zadeh is Professor in the Graduate School and director, Berkeley
initiative in Soft Computing (BISC), Computer Science Division and the
Electronics Research Laboratory, Department of EECs, Univeristy of California,
Berkeley, CA 94720-1776; Telephone: 510-642-4959; Fax: 510-642-1712;E-Mail:
zadeh@cs.berkeley.edu. Research supported in part by ONR Contract
N00014-99-C-0298, NASAContract NCC2-1006, NASA Grant NAC2-117, ONR Grant
N00014-96-1-0556, ONR Grant FDN0014991035, ARO Grant DAAH 04-961-0341 and the
BISC Program of UC Berkeley

```--
Dr. Masoud NikRavesh
Research Engineer - BT Senior Research Fellow
Chair: BISC Special Interest Group on Fuzzy Logic and Internet
Visiting Scientist: Lawrence Berkeley National Lab,
(Imaging and Collaborative Computing Group)
Berkeley initiative in Soft Computing (BISC)
Computer Science Division- Department of EECS
University of California, Berkeley, CA 94720
Phone: (510) 643-4522 - Fax: (510) 642-5775
Email: nikravesh@cs.berkeley.edu
URL: http://www.cs.berkeley.edu/~nikraves/
URL: http://www-bisc.cs.berkeley.edu/
URL: http://vision.lbl.gov/
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