Rational comparison of probabilities via a blow-up conjugacy

Abstract: "We consider the problem of producing a subjectively acceptable quantitative measure of the difference between two numerical measurements taking values between -1 and +1. We begin by listing the required properties of any such subjective difference measure. Then we define a family of measures via conjugacy in terms of a blow-up transformation ╬▓ which we initially leave unspecified, and describe the blown-up metric corresponding to a given ╬▓. We establish necessary and sufficient conditions on ╬▓ in order for the associated subjective difference measure to satisfy the properties of an admissible measure described previously, and we give several concrete choices for ╬▓ leading to admissible measures. We show that one of these measures has interesting connections with probability, Dempster-Shafer evidence theory, and special relativity. We show how to find a blow-up transformation ╬▓ leading to a given measure if one exists, thus in particular establishing the equivalence of the basic objects of our theory, namely 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."