text_combined: max makatchev dzvfuhitugmailcom dear lotfi seems one approach could based confidence interval semantics imprecise probabilities consider interval case confidence interval value aij corresponding distribution relative frequency dij event ij confidence interval probability union events ij kj based distribution dij dkj mean distribution sum means dij dkj coincides intuition sum intervals centered aijakj frequencies dij dkj independent variance dij dkj would equal sum variances fact variance sum less sum variances inverse dependency frequencies dij dkj covariance negative semantics would account following extreme case consider single random variable x taking values 1 2 n imprecise probability x1 xn interval marginal imprecise probability event x1 x2 x3 xn would interval zero length centered 1 confidence marginal probability equal 1 100 variance sum relative frequencies 0 regards maxim makatchev lotfi zadeh wrote x random variables taking values set 1 2 n entries joint probability matrix p form approximately aij aij take values unit interval add unity marginal probability distribution x two special cases approximately aij interpreted interval centering aij b approximately aij interpreted fuzzy triangular number centering aij warm regards lotfi uai mailing list wed 06 aug 2008 143105 0400 uai test problem involving imprecise probabilities