Lotfi -
> The standard axiomatic structure of standard probability theory
>does not address two basic issues which show their ungainly faces in
>many real-world applications of probability theory.They are (a)
>imprecision of probabilities;
Imprecision of probabilities can be handled with second order
probabilities and / or interval probabilities.
>and (b) imprecision of events.
If by this you mean representing events that don't satisfy the
clarity test, you are right. Probabilities are defined on sets -- or
in your terminology, "crisp sets." (I prefer to use standard
mathematical terminology unless there is a strong reason to do
otherwise. Therefore, I use"fuzzy set" to refer to the
generalization of standard set theory that you describe in your
email.)
>To address
>these issues and to add to probability theory the capability to deal
>with perception-based information, e.g., "usually Robert returns from
>work at about 6 pm; what is the probability that Robert is home at 6:30
>pm?"
I don't understand what you mean by calling this "perception based
information." This is a statement in natural language, at the
cognitive level and NOT at the perceptual level. I suspect we as a
community will ultimately conclude it is a dead-end path to try to
build a truly intelligent reasoner by reasoning directly with
linguistic statements such as this, using theories that are not based
on a deeper understanding of the complex process by which the brain
generates such linguistic statements. When we have a deeper
understanding of how we actually do cognitive level processing and
how it interacts with sub-symbolic processing, I doubt very much that
t-norms and t-conorms will turn out to be what it's about. They
strike me as epicycles.
I could be very wrong about that. I urge anyone who disagrees with
me to plunge in and work on t-norms and t-conorms. Science moves
forward because passionate adherents of the various theories try with
everything they've got to make their theories work. For what it's
worth, though, although I have the very highest respect for your
work, I'm placing my bets elsewhere.
In my mind, "perception based information" would mean something like
nerve impulses resulting from an optical or auditory waveform
impacting on the retina or the eardrum. If we are talking
automation, then a radar waveform or a bunch of pixel intensities
would qualify as "perception based information." It takes a huge
amount of processing to turn this kind of information into a
high-level linguistic summary, fuzzy or otherwise. We don't yet
understand very well how the brain does this, and our computational
models still reflect this lack of understanding -- although things
are moving very rapidly.
I am very keenly interested in figuring out how a brain or computer
can get from a bunch of pixels to a linguistic statement such as
"that is a roughly circular shaped blob." In my view, the big payoff
will come not from applying t-norms and t-conorms to reason about
statements already processed into linguistic form. The big payoff
will come from going from the sensory data to the linguistic
constructs. Once we understand how to do that, then how to reason
with the linguistic constructs will become obvious. It will fall
right out of the theory.
In my view, the most promising path to dealing with imprecision is
the following research program (which is going on at an active pace
as we speak):
- develop theories of how animals process sensory information;
- develop theories of how these sensory processing mechanisms
contribute to the generation of linguistic summary statements such as
the above;
- abstract away the details to arrive at the essential principles
of how humans and other animals do this;
- build computational theories that apply these essential principles.
In my view, it is quite likely that probability and decision theory
will prove adequate to the job. However, in my view it is also quite
likely that classical physics and classical computing will prove
inadequate to the job. We will need a new theory of computing in
which standard recursive function theory / Turing computability is
replaced by a quantum computation model that has quantum non-locality
and quantum randomness built into the computational apparatus from
the ground up. You can find some (as yet unpublished -- I haven't
yet written anything that I feel is ready for archival publication)
musings on this topic on my web site (url below). I am currently
working on a paper amplifying these ideas and developing a quantum
computation version of the physical symbol system hypothesis. Stay
tuned.
>It is necessary to generalize probability theory
Your preferred research program is to attack these problems by
generalizing probability theory via t-norms and t-conorms. That is a
perfectly valid approach to try, as I said above, although it is not
my preferred approach. However, it's not fair to say this is
NECESSARY unless you have definitively falsified the alternative
approaches. You demonstrate necessity only if you can show that
there is no way to handle these kinds of problems using ordinary
probability and decision theory. I believe you have come nowhere
near doing this.
>in three stages. A
>preliminary account of such generalization is described in a forthcoming
>paper of mine in the Journal of Statistical Planning and Inference,
>"Toward a Perception-based Theory of Probabilistic Reasoning with
>Imprecise Probabilities."
Thanks for the reference. I recommend it to anyone having a serious
interest in these issues. It is important to look at all the
alternatives before making up one's mind.
Respectfully,
Kathy Laskey
