S-Independence and S-Conditional Independence with respect to Upper and Lower Conditional Probabilities Assigned by Hausdorff Outer and Inner Measures

In this paper the notion of s-irrelevance with respect to upper and lower conditional probabilities assigned by Hausdorff outer and inner measures is proved to be a sufficient condition for strong independence introduced for credal sets. An example is given to show that the converse is not true. Moreover the definition of s-conditional irrelevance is given and a generalized factorization property is proposed as necessary condition of s-conditional irrelevance. Examples are given to show that s-conditional irrelevance and s-irrelevance are not related; moreover sufficient conditions are given for equivalence between s-conditional irrelevance and s-irrelevance. Finally the notion of s-irrelevance is extended to random variables.