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From: mackw@bytex.com
Subject: Re: Fuzzy theory or probability theory? 
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In article <94Nov29.133132edt.774@neuron.ai.toronto.edu>, 
<radford@cs.toronto.edu> writes:
> To take an actual real-world situation, consider the problem of a 
> police officer who has interviewed a witness to a crime.  The witness
> described the perpetrator as "tall".  How should we formalize how the
> officer should handle this information?
> 
> I think it is clear that what the police officer needs here is a 
> likelihood function - for any actual height (plus any other 
characteristics
> that might influence subjective assessment of height), the officer
> needs to know the PROBABILITY that the witness would describe the
> such a perpetrator as "tall".

Several comments regarding  "real world" situations (although some who 
know me may claim I am not qualified to comment on the real world).

1) Correct me if I'm wrong, but I don't believe that any probability 
measure of what heights are considered tall exists or are in use by 
police departments.  This would seem to contradict the statement that
the police officer needs this information to understand the description 
of "tall."

2) In reference to a different post, we need to remember that english and 
fuzzy logic equations are two different languages that we must translate 
between.  Thus the english language statement "tall and not tall" is not 
equivalent to the fuzzy language statement "tall AND NOT tall."  The 
correct translation of the fuzzy statement is "tall to a degree and not 
tall to another degree."  Literal, word for word translation, from fuzzy 
to english is no more valid than for any other language translation and 
can easily lead to confusion when we forget we are doing a translation 
step.

3) It is quite easy to restate a problem from terms of "membership" to 
terms of "probability."  An item's membership of a certain characteristic 
is related to the probablity that this characteristic will cause 
something to happen or not happen.  Restating the problem from one domain 
to another merely shows the problem statement exists in both domains; it 
does not imply which solution domain (fuzzy logic or probability) should 
be used.

4) Fuzzy logic does not concern itself very much with how the membership 
set equation is determined, probability is extremely concerned with 
determining this shape.  Fuzzy logic assumes that if a membership set 
definition is reasonably close, then the result will also be reasonably 
close.  It may also be noted that sampling methods also do not provide a 
precise membership set definition and usually includes an approximation 
of the likelihood that the definition is wrong.  Furthermore, the 
practice of curve fitting is itself a fuzzy operation; normal 
distributions do not exist in the real world (this is *not* saying they 
are not useful, I'm just trying to say they are approximations).

5) In most cases, it is *possible* to use probability to determine 
membership functions and the degree to which a specific characteristic 
contributes to a certain result, but it is not always *feasible*.  There 
are several reasons for this.  One, there may be too many variables to 
fully evaluate each one.  In the policeman example, consider how many 
possible descriptive characteristics a witness could use.  We are largely 
unwilling to expend the time, money, and effort to do a full statistical 
analysis of characteristic, but we still want our policeman to make use 
of this information.  Two, characteristics often are interactive, which 
would also require more probability tables.  For example, the probability 
that a witness would use the description "tall" is based somewhat on the 
height of the witness.  Three, when multiple characteristics contribute 
to a result, it is very difficult to work back from the probability of 
the result to determine the contribution of the individual 
characteristics.  Four, many times we need to know the result (or a good 
estimate) before we have the item to test.  If I'm going to manufacture 
red, sperical beads, I often need to  set the tolerances for redness and 
"spericalness" before I begin manufacturing them and can determine the 
probability of failure based on variations in these characteristics.  
(Aside: the "roundness" of a sphere is a difficult concept to define; 
lack of burrs, lack of pits, lack of corners, smoothness of curve, 
equivalence of various axes, etc).

In summary, if probability information is available or easily attainable, 
use it to determine membership sets.  The advantage of fuzzy logic is in 
cases where you do not want to obtain this data to this level of 
precision, you do not have to give up and say the problem is impossible 
to solve.  This why products are appearing using fuzzy logic which were 
not made using probability.  They  probably were possible using 
probability, it just wasn't worth the effort.

Wayne Mack
 
