**Computation with Imprecise
Probabilities—A Bridge to Reality**

Extended Abstract

An imprecise
probability distribution is an instance of second-order uncertainty, that is, uncertainty
about uncertainty, or uncertainty^{2} for short. Another instance is an
imprecise possibility distribution. Computation with imprecise
probabilities is not an academic exercise—it is a bridge to reality. In
the real world, imprecise probabilities are the norm rather than
exception. In large measure, real-world probabilities are perceptions of
likelihood. Perceptions are intrinsically imprecise, reflecting the bounded
ability of human sensory organs, and ultimately the brain, to resolve detail
and store information. Imprecision of perceptions is passed on to perceived
probabilities. This is why real-world probabilities are, for the most part,
imprecise.

What is important to
note is that in applications of probability theory in such fields as risk assessment,
forecasting, planning, assessment of causality and fault diagnosis, it is a
common practice to ignore imprecision of probabilities. The problem with this
practice is that it leads to results whose validity is in doubt. This
underscores the need for approaches in which imprecise probabilities are
treated as imprecise probabilities rather than as precise probabilities.

Peter Walley's
seminal work "Statistical Reasoning with Imprecise Probabilities,"
published in l99l, sparked a rapid growth of interest in imprecise
probabilities. Today, we see a substantive literature, conferences,
workshops and summer schools. An exposition of mainstream approaches to
imprecise probabilities may be found in the 2002 special issue of the Journal
of Statistical Planning and Inference (JSPI), edited by Jean-Marc Bernard. My
paper "A
perception-based theory of probabilistic reasoning with imprecise probabilities,"
is contained in this issue but is not a part of the mainstream. A
mathematically rigorous treatment of elicitation of imprecise
probabilities may be found in "A behavioural model for vague probability
assessments," by Bert de Cooman, Fuzzy Sets and Systems, 2005.

The approach which
is outlined in the following is rooted in my l975 paper "The
concept of a linguistic variable and its application to approximate reasoning,"
Information Sciences, but in spirit it is close to my 2002 JSPI paper.
The approach is a radical departure from the mainstream. Its principal
distinguishing features are: (a) imprecise probabilities are dealt with
not in isolation, as in the mainstream approaches, but in an environment of
imprecision of events, relations and constraints; (b) imprecise probabilities
are assumed to be described in a natural language. This assumption is
consistent with the fact that a natural language is basically a system for
describing perceptions.

The capability to
compute with information described in a natural language opens the door to
consideration of problems which are not well-posed mathematically. Following
are very simple examples of such problems.

*X*is a real-valued random variable. What is known about*X*is: (a) usually*X*is much larger than approximately*a*; and (b) usually*X*is much smaller than approximately*b*, with*a*<*b*. What is the expected value of*X*?*X*is a real-valued random variable. What is known is that Prob(*X*is small) is low; Prob(*X*is medium) is high; and Prob(*X*is large) is low. What is the expected value of*X*?- A box contains approximately
twenty balls of various sizes. Most are small. There are many more
small balls than large balls. What is the probability that a ball drawn at
random is neither large nor small?
- I am checking-in for my flight. I ask
the ticket agent: What is the probability that my flight will be
delayed. He tells me: Usually most flights leave on time.
Rarely most flights are delayed. How should I use this information
to assess the probability that my flight may be delayed?

To compute with
information described in natural language we employ the formalism of Computing
with Words (CW) (Zadeh l999) or, more generally, NL-Computation (Zadeh 2006). The
formalism of Computing with Words, in application to computation with
information described in a natural language, involves two basic steps: (a)
precisiation of meaning of propositions expressed in natural language; and (b)
computation with precisiated propositions. Precisiation of meaning is achieved
through the use of generalized-constraint-based semantics, or GCS, for
short. The concept of a generalized constraint is the centerpiece of GCS.
Importantly, generalized constraints, in contrast to standard constraints, have
elasticity. What this implies is that in GCS everything is or is allowed to be
graduated, that is, be a matter of degree. Furthermore, in GCS everything is or
is allowed to be granulated. Granulation involves partitioning of an
object into granules, with a granule being a clump of elements drawn together
by indistinguishability, equivalence, similarity, proximity or functionality.

A generalized constraint
is an expression of the form *X *isr *R*, where *X* is the constrained variable, *R*
is the constraining relation and *r* is
an indexical variable which defines the modality of the constraint, that is,
its semantics. The principal modalities are: possibilistic (*r* = blank), probabilistic (*r* = *p*),
veristic (*r* = *v*), usuality (*r *= *u*) and group (*r *= *g*).
The primary constraints are possibilistic, probabilistic and veristic. The
standard constraints are bivalent possibilistic, probabilistic and bivalent veristic.
In large measure, scientific theories are based on standard constraints.

Generalized
constraints may be combined, projected, qualified, propagated and
counterpropagated. The set of all generalized constraints, together with
the rules which govern generation of generalized constraints from other
generalized constraints, constitute the Generalized Constraint Language
(GCL). Actually, GCL is more than a language—it is a language
system. A language has descriptive capability. A language
system has descriptive capability as well as deductive capability.
GCL has both capabilities.

The concept of a
generalized constraint plays a key role in GCS. Specifically, it serves two
major functions. First, as a means of representing the meaning of a proposition,
*p*, as a generalized
constraint; and second, through representation of *p* as a generalized constraint it
serves as a means of dealing with *p*
as an object of computation. It should be noted that
representing the meaning of *p* as
a generalized constraint is equivalent to precisiation of *p* through translation into
GCL. In this sense, GCL plays the role of a meaning precisiation
language. More importantly, GCL provides a basis for computation with
information described in a natural language. This is the province of CW or,
more generally, NL-Computation.

A concept which
plays an important role in computation with information described in a natural
language is that of a granular value. Specifically, let *X* be a variable taking values in a
space *U*. A granular value of *X*, **u*,
is defined by a proposition, *p*, or
more generally by a system of propositions drawn from a natural
language. Assume that the meaning of *p* is precisiated by representing it as a generalized constraint,
GC(*p*). GC(*p*) may be viewed as a definition of the granular value, **u*. For example, granular values
of probability may be defined as approximately 0.l, ..., approximately
0.9, approximately l. A granular variable is a variable which takes granular
values. For example, young, middle-aged and old are granular values of
the granular variable Age. The probability distribution in Example 2 is an
instance of a granular probability distribution. In effect, computation
with imprecise probability distributions may be viewed as an instance of
computation with granular probability distributions.

In the CW-based
approach to computation with imprecise probabilities, computation with
imprecise probabilities reduces to computation with generalized
constraints. What is used for this purpose is the machinery of GCL. More
specifically, computation is carried out through the use of rules which govern
propagation and counterpropagation of generalized constraints. The principal
rule is the extension principle (Zadeh l965, l975).
In its general form, the extension principle is a computational schema which
relates to the following problem. Assume that *Y* is a given function of *X*,
*Y* = *g*(*X*). Let **g* and **X* be granular values of *g* and
*X*, respectively. Compute **g*(**X*).

In most computations
involving imprecise probabilities what is sufficient is a special form of the
extension principle which relates to possibilistic constraints. More
specifically, assume that *f* is a
given function and *f*(*X*) is constrained by a possibility
distribution, *A*. Assume that *g* is a given function, *g*(*X*).
The problem is to compute the possibility distribution of *g*(*X*)
given the possibility distribution of *f*(*X*). In this case, the extension
principle reduces the solution of the problem in question to the solution
of a variational problem (Zadeh 2006).

In summary, the
CW-based approach to computation with imprecise probabilities opens the door to
computation with probabilities, events, relations and constraints which are
described in a natural language. Progression from computation with
precise probabilities, precise events, precise relations and precise
constraints to computation with imprecise probabilities, imprecise events,
imprecise relations and imprecise constraints is an important step forward—a step
which has the potential for a significant enhancement of the role of natural
languages in human-centric fields such as economics, decision analysis,
operations research, law and medicine, among others.

**Neuroeconomics:
yet another field where rough sets can be useful?**

Janusz
Kacprzyk, Fellow of IEEE

Systems
Research Institute, Polish Academy of Sciences

ul.
Newelska 6, 01–447 Warsaw, Poland

E-mail:
kacprzyk@ibspan.waw.pl

WWW: www.ibspan.waw.pl/
kacprzyk

Google:
kacprzyk

**Abstract**

We
deal with neuroeconomics which may be viewed as a new emerging field of
research at the crossroads of economics, or decision making, and brain
research. Neuroeconomics is basically about neural mechanisms involved in
decision making and their economic relations and connotations. We briefly
review first the traditional formal approach to decision making, then discuss
some experiments of real life decision making processes and point our when and
where the results prescribed by the traditional formal models are not
confirmed. We deal with both decision analytic and game theoretic type models.
Then, we discuss results of brain investigations which indicate which parts of
the brain are activated while performing some decision making related courses
of action and provide some explanation about possible causes of discrepancies
between the results of formal models and experiments. We point out the role of
brain segmentation techniques to determine the activation of particular parts
of the brain, and point out that the use of some rough sets approaches to brain
segmentation, notably by Hassanien, ´Sl¸ezak and their collaborators, can
provide useful and effective tool.

**Research Directions in the KES Centre**

Lakhmi
Jain

School
of Electrical and Information Engineering,

Knowledge
Based Intelligent Engineering Systems Centre,

University
of South Australia, Mawson Lakes, SA 5095, Australia.

Lakhmi.Jain@unisa.edu.au

**Abstract**

The ongoing success of the Knowledge-Based Intelligent
Information and Engineering Systems (KES) Centre has been stimulated via
collaborated with industry and academia for many years. This Centre currently
has adjunct personnel and advisors that mentor or collaborate with its students
and staff from Defence Science and Technology Organisation (DSTO), BAE Systems
(BAE), Boeing Australia Limited (BAL), Ratheon, Tenix, the University of
Brighton, University of the West of Scotland, Loyola College in Maryland,
University of Milano, Oxford University, Old Dominion University and University
of Science Malaysia. Much of our research remains unpublished in the public
domain due to these links and intellectual property rights. The list provided
is non-exclusive and due to the diverse selection of research activities, only
those relating to Intelligent Agent developments are presented.

[1] Dedicated to Peter Walley.

* Department
of EECS,

E-Mail: zadeh@eecs.berkeley.edu . Research supported in part by ONR N00014-02-1-0294, BT Grant CT1080028046, Omron Grant, Tekes Grant, Chevron Texaco Grant and the BISC Program of UC Berkeley.