Re: Probability and possibility


Subject: Re: Probability and possibility
From: Gert de Cooman (Gert.DeCooman@rug.ac.be)
Date: Fri Nov 10 2000 - 17:35:55 MET


> I would like to get a clear explanation of the difference between possiblity
> measure and probability measure. In other words, I would like to know the
> answer to the question: "What possibility can do that probability cannot".

I think there are a number of ways in which this question could be
answered, depending on the interpretation that is given to both the
probability and the possibility measures. I will only discuss the case
here that these are interpreted as epistemic uncertainty models, that
is, models reflecting the "knowledge" or "beliefs" of an agent. In this
case, there is a standard argument by Bayesians (originating in de
Finetti's and Ramsey's work) called a Dutch book argument, claiming to
show that any rational agent should model his beliefs using a
probability measure. There is one questionable assumption in this
argument, stating essentially that a rational agent, whatever his
knowledge is, should always prefer one of two given options. He is not
allowed to refrain from making a choice between them, because for
instance he would not have not enough information to make an informed
choice (there is a difference between choice and preference, preference
implies choice, but a choice can be arbitary and not based on
preference, e.g. if it is enforced as in the Bayesian argument). If this
questionable rationality requirement is removed, and the other
rationality requirements are kept, we find that an agent's beliefs
should be modeled not necessarily by a probability measure, but by what
is called a coherent (i.e. rational) *imprecise* probability model. A
specific example of such a model is a coherent upper probability, and
normal possibility measures are specific examples of such coherent upper
probabilities (as are probability measures, of course). So in this view,
possibility measures and probability measures are just special cases of
a much more general class of models that are available for modelling
beliefs and knowledge. The more information is available, the more
choices an agent will be willing to actually make, and the more precise
(i.e. the closer to a probability measure) his model becomes.
Possibility measures are uncertainty models that are fairly imprecise,
meaning that they reflect weak information states. It has been argued
that the information conveyed by statements in natural language is
fairly weak, and can be modelled by possibility measures.

You can find more information on imprecise probability models in the web
site http://ippserv.rug.ac.be.
More information on the connection between possibility measures and
(imprecise) probability theory can also be found in a few of my papers,
cited below. Many of these can be downloaded form my web site
(ippserv.rug.ac.be/~gert)

I hope this clarifies at least one side of this issue.

Best wishes,

Gert de Cooman

Gert de Cooman and Dirk Aeyels, A random set description of a
possibility measure and its natural extension, 6 pages, accepted for
publication in IEEE Transactions on Systems, Man and Cybernetics, 1999.

Gert de Cooman and Dirk Aeyels, Supremum preserving upper probabilities,
Information Sciences, 1999, vol. 118, pp. 173-212.

Peter Walley and Gert de Cooman, Coherence of rules for defining
conditional possibility, International Journal of Approximate Reasoning,
1999, vol. 21, pp. 63-107.

Gert de Cooman and Peter Walley, An imprecise hierarchical model for
behaviour under uncertainty, 34 pages, submitted for publication in
Theory and Decision, 1999

Gert de Cooman, Integration and conditioning in numerical possibility
theory, 29 pages, submitted for publication in Annals of Mathematics and
Artificial Intelligence, 1999.

-- 
==============================================================
Prof. dr. ir. Gert de Cooman
Onzekerheidsmodellering en systeemtheorie/
Uncertainty modelling and systems theory
--------------------------------------------------------------
E-mail: gert.decooman@rug.ac.be
URL:    http://ippserv.rug.ac.be/~gert
--------------------------------------------------------------
Universiteit Gent
Onderzoeksgroep SYSTeMS
Technologiepark - Zwijnaarde 9
9052 Zwijnaarde 
Belgium
--------------------------------------------------------------
Tel: +32-(0)9-264 56 53
Fax: +32-(0)9-264 58 40
==============================================================

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