University of California, Los Angeles, 520 Portola Plaza, Los Angeles, CA 90095, USA.
The aim of this work is further advancement of hyperprobability theory by including negative hyperprobabilities in the scope of this theory and building an axiomatic theory of symmetric hyperprobabilities. Hyperprobabilities are more general than conventional probabilities being constructed using real hypernumbers instead of real numbers. This extension allows essential extension of the scope of probability theory and its applications. Another extension of probability theory is related to negative probabilities. Although classical probability is a function with values in the interval [0, 1], researchers found that negative probabilities could be a useful tool in physics, economics and some other areas. In this paper, we synthesize both approaches developing theory of symmetric hyperprobabilities, which can take both positive and negative values in hypernumbers. In Section 2, some constructions and concepts from the theory of hypernumbers and extrafunctions are presented. Section 3 develops foundations for symmetric hyperprobabilities. In Section 4, symmetric hyperprobabilities are introduced and studied.
Key words: Negative probability, Hyperprobability, Annihilation, Symmetry, Axiom.