Two generalizations of the semi-graphoid rule of probabilistic independence and more

Probabilistic independence is a key concept in probability theory and statistics. For probabilistic independence a set of well known qualitative rules exists, the so-called semi-graphoid rules, which can be summarized into a single semi-graphoid rule. This rule system was conjectured to be complete, it is however incomplete and an additional five rules were formulated. The generalization of one of those rules subsequently showed that no finite rule system exists and in recent work even all five additional rules were (further) generalized. In this paper, two new generalized rules are stated, both involving n, variable sets . These rules generalize the semi-graphoid rule for n is odd and generalize one of the additional rules for n is even. Furthermore two new rules of probabilistic independence are given. The paper thereby contributes to the insights into the structural properties of probabilistic independence and provides an enhanced description of probabilistic independence by means of rules.