Re: Thomas' Fuzziness and Probability

From: Herman Rubin (hrubin@odds.stat.purdue.edu)
Date: Thu Aug 23 2001 - 08:13:45 MET DST

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In article <9lrutb\$8cn\$2@fbi-news.cs.uni-dortmund.de>,
Stephan Lehmke <Stephan.Lehmke@cs.uni-dortmund.de> wrote:
>In article <m28zgeg6zd.fsf@pusch.xnet.com>, Gordon D. Pusch writes:
>> Stephan.Lehmke@cs.uni-dortmund.de (Stephan Lehmke) writes:

>>> Btw, I doubt that the fact that you know something is reflected by
>>> probabilistic logic, without your formalizing it in any way.

>> That is =EXACTLY= what Bayesian probability theory is all about !!!
>> In Bayesian probability theory, _ALL_ probabilities are conditional
>> on the knowledge base one is willing to apply to the problem at hand.
>> The conditional probability P(A|{B}) in Bayesian theory is the degree
>> of confidence one has in the truth of Boolean proposition 'A', given
>> that the set of Boolean propositions {B} (the knowledge base one is
>> willing to apply to the problem) is assumed to be true. The Laws of
>> probability and Bayes theorem provide all the tools one needs to reason
>> about such `uncertain' Boolean propositions, as well as to incorporate
>> new data into one's knowledge base.

[...]

>>> Of course, it is perfectly possible to state the relations between A
>>> and B it the form of axioms, as I have pointed out before.

>> ...And that is what all Bayesian probabilities are conditioned on: The set
>> of Boolean popositions one is willing to take as axiomatic and relevant to
>> the problem at hand.

>I hate to burst your bubble, but that's exactly the same in _any_ form
>of knowledge representation by logical formulae. Whether the
>underlying logic is classical, fuzzy, probabilistic, possibilistic,
>bayesian, dempster/shafer or any other of the myriad variants and/or
>combinations might change the semantics of computed degrees associated
>with derived formulae, but it doesn't change in the least the general
>ability of the system to deal with background knowledge.

>So the question how well background knowledge is handeled is
>completely unsuitable for distinguishing between probabilistic and
>fuzzy logic.

Variants, such as Dempster/Shafer and fuzzy, do not give
answers as to what ACTION to take.

BTW, it is not necessary that all probabilities are conditional,
and combinations of probabilities of various types give no
problems whatever.

Further, the behavioral Bayesian approach is not based on
beliefs. It is that the utility function in ignorance of
the state of nature is a positive linear functional (or
integral with respect to a "prior") of the utility functions
for the various states of nature. This is an important
distinction, as it states that to get robust approximations
the integral needs to be approximated, not the measure.
These are quite different even in relatively simple problems.

```--
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
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