Non-Archimedean Probabilities and Non-Archimedean Bayesian Networks

In the paper we consider non-Archimedean fuzziness and probabilities. The idea of non-Archimedean multiple-validities is that (1) the set of values for the vagueness and probability is uncountable infinite and (2) this set is not wellordered. For the first time the non-Archimedean logical multiple-validity was proposed in [13], [14]. We propose non-Archimedean fuzziness that is defined on an infinite-order class of fuzzy subsets in the framework of infinite-order (ω-order) vagueness. This approach allows to set an ω-order fuzzy logic such that its well-formed formulas have truth values in an interval [0, 1] of hyperreal or hyperrational numbers. On the base of non-Archimedean fuzzy logic we can build also nonArchimedean probability logic. In this paper I propose to define probabilities an algebra of fuzzy subsets. These probabilities are said to be fuzzy ones. Their main originality consists in that some Bayes’ formulas do not hold in the general case. In the framework of ω-order vagueness we can construct infinitely hierarchical Bayesian networks. For instance, we can consider i-order variables of Bayesian network as i-tuples of first-order variables and ω-order variables as infinite tuples of first-order variables. Also, for the first time we propose to use infinite-order logical constructions for setting fuzziness and probabilities. Let us remember that Archimedes’ axiom affirms: for any positive real or rational number ε, there exists a positive integer n such that ε ≥ 1 n or n · ε ≥ 1. The field that satisfies all properties of R without Archimedes’ axiom is called the field of hyperreal numbers and it is denoted by ∗R. The field that satisfies all properties of Q without Archimedes’ axiom is called the field of hyperrational numbers and it is denoted by ∗Q. By definition of field, if ε ∈ R (resp. ε ∈ Q), then 1/ε ∈ R (resp. 1/ε ∈ Q). Therefore ∗R and ∗Q contain simultaneously infinitesimals and infinitely large integers: for an infinitesimal ε, we have N = 1ε , where N is an infinitely large integer. In the standard way, probabilities are defined on an algebra of subsets. Recall that an algebra A of subsets A ⊂ X consists of the following: (1) union, intersection, and difference of two subsets of X; (2) ∅ and X. Then a finitely additive probability measure is a nonnegative set function P(·) defined for sets A ∈ A that satisfies the following properties: