# RE: BISC: A Challenge to Bayesians

Subject: RE: BISC: A Challenge to Bayesians
From: Martin Lefley (mlefley@bournemouth.ac.uk)
Date: Sat Aug 05 2000 - 07:54:49 MET DST

All,

I am not a Bayesian or a Fuzzarian(?) though I tend towards using fuzzy
logic for its simplicity and elegance. There is a hint in this message that
Bayesian and Probability theory can not deal with some situations so I feel
this should be addressed. In fact Lofti does show how by extending
probability theory such concepts can be incorporated, one could also see
this as en extension of fuzzy logic which I think fuzzy researchers should
be aware of.

Probability estimates have validity in all contexts, as does fuzzy logic,
given the amount of data. If you take single statements and extend reasoning
about them in isolation then you make an erroneous extrapolation. In fact
the only way to extend reasoning is by experience of the world, i.e. other
similar events.

The only way to extrapolate from a statement such as

> --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?
>
is by knowledge of other human behaviours. Fuzzy logic tends to extrapolate
by general interpretation of language, e.g. the general usage of "earliest"
and "high" above. There are problems with this generalisation e.g the recent
discussion on this list of the meaning of "tall" in different contexts
(probably best dealt with by carefully defining the relevant question).
Probability methods would extrapolate from the distribution of arrivals of
e.g. people generally (probably Poisson distribution) and use this to make
an estimate of P(Robert home) > T.

I don't mind healthy competition but all researchers should consider all
approaches and use any data and paradigm that can help solve a problem. Each
has strengths and weaknesses and the classic trade off in AI is between
generalisation and focus, whatever the paradigm. Probably the biggest
problem with Probability theory is intractability and computing effort,
approximations and more powerful computers are helping to reduce this.

Martin
> ----------
> From: Michelle T. Lin[SMTP:michlin@eecs.berkeley.edu]
> Sent: Sunday, July 30, 2000 8:50 PM
> To: Multiple recipients of list
> Subject: BISC: A Challenge to Bayesians
>
> *********************************************************************
> Berkeley Initiative in Soft Computing (BISC)
> *********************************************************************
>
>
> To: BISC Group
>
>
> 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
> 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
>
> ----------------------------------------------------------
> 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
> ----------------------------------------------------------
> me(zadeh@cs.berkeley.edu) with cc to Michael Berthold
> (berthold@cs.berkeley.edu)
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