We introduce the notion of a probabilistic measure which takes values in hyperbolic numbers and which satisfies the system of axioms generalizing directly Kolmogorov’s system of axioms. We show that this new measure verifies the usual properties of a probability; in particular, we treat the conditional hyperbolic probability and we prove the hyperbolic analogues of the multiplication theorem, of the law of total probability and of Bayes’ theorem. Our probability may take values which are zero–divisors and we discuss carefully this peculiarity.
Alpay, D., Luna-Elizarrarás, M.E., Shapiro, M., 2017. Kolmogorov’s Axioms for Probabilities with Values in Hyperbolic Numbers. Advances in Applied Clifford Algebras 27, 913–929. doi:10.1007/s00006-016-0706-6