Re: Imprecise Probability


Subject: Re: Imprecise Probability
From: Breezy (weiz@spam-block.sympac.com.au)
Date: Fri Nov 24 2000 - 03:28:38 MET


G'day

and sorry because I am wordy tonight.

That you want to convert a number (even a probability measure) into a
linguistic measure
is not in its self a reason to deploy fuzzy logic.

You could of course simply define a series of arbitrary points on the
universe of all your probabilities.
Below a point it is linguistically described as low probability.

So for a range of weights 1 kilo to 10 kilo let's say under 3 is light. 3-7
is medium and 7 + is heavy.

Nothing fuzzy yet for either probability or weights.

Now to fuzzy things up, is to rate the extent that your probability measure
or my weight measures
belong to their respective linguistic grouping. This grouping is to be known
as a fuzzy set.

1 Kilo is very much a part of the light fuzzy set, and only the light fuzzy
set. It is therefore a high value member.

3 kilo is less a member of the light fuzzy set cause it could also fit as a
member of the medium weight set.

3 kilo in fuzzy terms is a low value member of both the light and medium
sets. Its validity in either set is denoted
by a membership value. This value typical is between 0 and 1, so the closer
to 1 the greater the membership value
and the greater the validity of the linguistic descriptor assigned it.

The fuzzying of the weight range has blurring that crisp arbitrary under 3
light and over 3 medium.

So to your probability scale. Probability may as well be kilos or bananas,
it's only a descriptor of a numeric universe.

0.78% probability on a scale that goes all the way to 100% probability is
most definitely in the low probability fuzzy set
It has a big membership value in that set.

One option for a fuzzy set layout on a scale of 0 to 100 would see that from
25% to 50% is both in low and medium probability
fuzzy sets. 49% probability (or 49 bananas out of 100) is a very weak member
of the low probability set and a very valid
member of the medium probability set.

Membership value of 49% in low probability is closer 0 and the membership
value of 49% in the medium probability fuzzy set
is much closer to 1.

So ??

 Well the rest is all about using the linguistic value as you do in every
day discourse but lying in the background is
a fuzzy mathematical value for each of the members of say the low
probability set.

let' do it *********** We have a low probability of rain if there is a
large amount of sunlight. ************

you & I have just talked about two fuzzy set spaces. probability and
sunlight.
"low probability" is one of the fuzzy sets in the total range (or universe)
of probabilities
and "large amount" is one of the fuzzy sets in the entire universe of all
sunlight measures.

So tell me how much sunlight there is and a good fuzzy logic predictive
modeling system will make an
informed guess about how much rain there will be.

But prediction is not just the name of the game.

The sentence above (now a fuzzy formula) is a knowledge transferal system.
Earthlings to machine type of communication
that makes sense to both. Even Machine to machine learning.

I can pass streams of raw data through a fuzzy based data mining system that
will tell me the relationships in that raw
data and deliver the knowledge in sentences like that above.

Those sentences or rules or knowledge nuggets (data mining speak) can be
used in a fuzzy predictive modeling system or
just read by us to give us greater insight into the particular system
producing the data.

If I'm rambling it's late :- another knowledge (rambling is a fuzzy measure
of the number of sentences I am using. late is time lapsed since waken) a
good fuzzy predictive system with the knowledge of the correlation between
number of sentences and time since awaken
would need only count the number of sentences in this email and know I have
been up too long. It might even tell us the time if it knew when I woke.

I always recommend reading Mr Earl Cox- The Fuzzy systems handbook for
further reading on all things fuzzy.
Earl has probably moved on and written many other fine things but this book
just did it for me.

Otto Cordero wrote in message
<003501c04e4c$ddfe85a0$22960ac8@cti.espol.edu.ec>...
>
>Hello
>
>I would like to be more specific about my previous question....
>Lets supose someone is asked to give his estimation about the probability
of
>some event, answers like "0.78 of probability" are very unlikely. I would
>expect something as "high probability" or "low probability", here we
>linguistic terms, wich could be associated with fuzzy sets.
>I am asking for more information on how fuzzy sets theory joins with
>probabilty theory to manage this kind of situations.
>
>Thanks a lot.
>
>Otto
>
>
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