Axiomatization of Implication for Probabilistic Independence and Unary Variants of Marginal Identity and Marginal Distribution Equivalence

We consider probabilistic independence and unary variants of marginal identity and marginal distribution equivalence over finite probability distributions. Two variables x and y satisfy a unary marginal identity when they are identically distributed. If the multisets of the marginal probabilities of all possible values for variables x and y are equal, the variables satisfy a unary marginal distribution equivalence. This paper offers a sound and complete finite axiomatization and a polynomial-time algorithm for the implication problem for probabilistic independence, unary marginal identity, and unary marginal distribution equivalence.