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
Wean Hall, 6th Floor
Carnegie Mellon University
Pittsburgh, PA 15213