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
    Professor Lotfi A. Zadeh
    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/ -------------------------------------------------------------------- If you ever want to remove yourself from this mailing list, you can send mail to <Majordomo@EECS.Berkeley.EDU> with the following command in the body of your email message: unsubscribe bisc-group or from another account, unsubscribe bisc-group <your_email_adress>

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