Rational comparison of probabilities
via a blow-up conjugacy
Sergio A. Alvarez
Center for Nonlinear Analysis and
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213-3890
Abstract
The problem of assigning a meaningful and useful value to the
difference between two numerical measurements is an important
one that arises in a variety of contexts. Examples include
the study of directional asymmetry in zoology and high-energy
physics, the measurement of lateralization in neurological studies
of handedness, language, and vision, and the combination of
degrees of belief and disbelief in uncertain reasoning and
knowledge-based systems.
In the present paper we address the above problem in a
systematic way. We list the desired properties of any
admissible subjective difference measure.
Then we define a family of measures via conjugacy relative to
a blow-up transformation b
and describe the blown-up metric associated with
a given choice for b.
We find necessary and sufficient conditions on
b so that the associated
difference measure is admissible, and describe several
possible choices for b.
We show how to find a blow-up transformation
b leading to a given measure
if one exists, thus in particular establishing the
equivalence of the basic objects of our theory:
the difference measure, the blow-up transformation,
and the metric.
We close by illustrating the application of one of our
measures to lateralization assessment in a computational
simulation of sensory map formation in a bihemispheric
brain.
Paper available as Research Report No. 97-NA-010
Center for Nonlinear Analysis
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