A transformation-based framework for
nonlinear combination operators
Annals of Mathematics and Artificial Intelligence,
40 (3/4), 187-213, March 2004
Sergio A. Alvarez
Computer Science Department
Boston College
Chestnut Hill, MA 02467
Abstract
Approximate reasoning in the presence of multiple sources of
information requires a mechanism to combine different measures
representing uncertainty, belief, or desirability into a single
aggregate measure that reflects contributions of the individual
information sources. This may be achieved by constructing an
appropriate combination operator. Specific combination operators
are provided by formalisms such as probability theory and
Dempster-Shafer evidence theory. Various ad-hoc approaches to
combination have also been proposed, a classical example being
the MYCIN calculus of certainty factors. In the present paper we
present an analytical theory of combination operators, focusing
on the normalizing frames of reference on the space in which the
measures to be combined take their values, which reduce certain
combination operators to the canonical arithmetic sum operator.
The cornerstone of our theory is an algorithm that is guaranteed
to explicitly construct a normalizing reference frame directly from
a given combination operator whenever such a frame exists.
This procedure includes a criterion to settle the frame
existence issue for a given combination operator.
Our approach provides a natural nonlinear scaling mechanism that
extends combination operators to parameterized families,
allowing one to adjust the sensitivity of these operators to new information.
We give a procedure to reconstruct a frame transformation from
the group of nonlinear scaling operations associated with it.
Furthermore, our theory allows prediction and control of the asymptotic
growth rate of the consensus belief values produced by combination
operators in the presence of a large number of information sources.