Re: PROBABILITY-POSSIBILITY CONSISTENCY

Ellen Hisdal (ellen@ifi.uio.no)
Mon, 15 Dec 1997 06:16:31 +0100 (MET)

On Mon, 8 Dec 1997 wsiler@aol.com ?? wrote:

> On Fri, Nov 28, Ellen Hisdal wrote:
>
> >Prob(ui)=P(ui|a)=probability that an object which has been labeled yes-a
> >by a subject has the (exactly measured) height ui.
> >
> >Poss(ui)=Prob(a|ui)=probability that a subject will assign to an object of
> >(exactly measured) height ui the label yes-a.
> >
> >
> I have no problem with the definition of Poss(ui)l. However, I do have a
> problem with the definition of Prob(ui). Here we are into probability with a
> continuous argument. Ordinarily, this would be described by a probability
> density function on ui, with the usual restriction of area = 1. The probability
> that any one exactly measured ui would have produced the label would be zero:
> the density function would be, of course, d(cumprob(ui|a)/dui. I think we could
> talk about the likelihood that a height labeled yes-a would have exactly the
> value ui, but that is a different story, and such a likelihood could range from
> zero to a maximum of infinity for a crisp number.
>
> Am I missing something?
>

One has to choose whether one wishes to work with a continuous universe
(and consequently with a probability density),
or with a `quantized' universe in connection with probabilities.
I chose the latter because it is somewhat easier to work with.
This is why I used the symbol `ui' (u with subscript `i')
for the argument of the probability function. Suppose that u denotes a
height value, and that the domain of ui consists of the height values
5cm, 15 cm, ... 165cm, 175cm,... .
Prob(ui=175cm) then actually stands for the probability that the height u
(of a person in a given population)
lies in the interval [170cm, 180cm). The sum of P(ui) over all ui's
must be equal to 1.

In contrast poss(u) (for tall) is a subject's estimate of the probability
that a person of height u will be labeled `tall' (by some other person).
In this case it does not make much difference
whether u is a continuous or a quantized variable.

The possibility of ui for `tall' should be denoted by
prob(tall|ui). It should be compared with prob(ui|tall), the probability
that a person labeled `tall' has the height ui.

The following formula connects the two probabilities
(i.e. prob(ui|tall) and (poss(ui) for `tall') = prob(tall|ui) ):

prob(ui|tall)=prob(ui) x prob(tall|ui) / prob(tall)

where

prob(tall)=sum over i of [prob(ui) x prob(tall|ui)]

(see reference [1], eqs. (22), (23) and reference [4] eq.).

In addition it is shown in the references below that one must
differentiate between the measured value of the height of a person
(denoted by u or ui here, and by u with superscript `ex' (for `exact') in the
references below), versus the estimate by an observer of the height
of an observed object (person). The observer is supposed to assign
either the label yes-tall or the label `no-tall' to the object.
This estimate is denoted by `u' in the references below.

In addition we have the person who is required to assign a membership
value in `tall' to an object of specified measured height. This
person takes account of various sources of uncertainty (see reference [2])
which may influence the decision of another person who is supposed to
choose between the labels `yes-tall' and `no-tall' for an object of a
given height (this person only estimates the heightof the object).
The possibility or grade-of-membership concept is actually quite
a complicated one when you analyse it in detail.

Greetings
Ellen Hisdal

References:
[1]
@article{inf1p1,
author = {Hisdal, E.},
title = {Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part~1.1. {D}ifficulties with Present-Day
Fuzzy Set Theory and their Resolution in the {TEE} Model},
journal = {Int. J. Man-Machine Studies},
year = {1986},
volume = {25},
pages = {89-111},
ignored = {page 94 for different words for grade of membership
page 95 for lack of difference between distr tall|u and u|tall},
}
[2]
@article{inf1p2,
author = {Hisdal, E.},
title = {Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part~1.2. {D}ifferent Sources of Fuzziness
and Uncertainty},
journal = {Int. J. Man-Machine Studies},
year = {1986},
volume = {25},
pages = {113-138} }

[3]
@techreport{inf1p3,
author = {Hisdal, E.},
title = {Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part~1.3. {R}eference Experiments and
Label Sets},
institution = {Institute of Informatics, University of Oslo, Box
1080 Blindern, 0316 Oslo 3, Norway},
year = {1988,1990},
type = {Research Report},
note = {ISBN~82-7368-053-3.
Can also be found on
http://www.ifi.uio.no/$\sim$ftp/publications/research-reports/Hisdal-3.ps},
number = {147} }

[4]
@techreport{inf1p4,
author = {Hisdal, E.},
title = {Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part~1.4. {T}he {TEE} Model},
institution = {Institute of Informatics, University of Oslo, Box
1080 Blindern, 0316 Oslo 3, Norway},
year = {1988,1990},
type = {Research Report},
note = {ISBN~82-7368-054-1.
Can also be found on
http://www.ifi.uio.no/$\sim$ftp/publications/research-reports/Hisdal-4.ps},
number = {148} }

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Ellen Hisdal | Email: ellen@ifi.uio.no
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