Dear Lotfi, It seems to me that one approach could be based on the confidence interval semantics of imprecise probabilities. That is, consider the interval in the case (a) as a confidence interval for the value of aij, corresponding to some distribution of the relative frequency dij of the event ij. Then the confidence interval for the probability of the union of events ij and kj should be based on the distribution of dij + dkj.
The mean of this distribution is the sum of the means of dij and dkj, which coincides with the intuition that the sum of the intervals should be centered at aij+akj. If frequencies dij and dkj were independent, then the variance of dij + dkj would be equal to the sum of the variances. But in fact the variance of the sum will be less than the sum of the variances because of the inverse dependency between frequencies dij and dkj (their covariance is negative). This semantics would account for the following extreme case. Consider a single random variable X taking values from (1, 2, .., n). While each imprecise probability of X=1, ..., X=n can be an interval, the "marginal" imprecise probability of the event (X=1, or X=2, or X=3,..., or X=n) would be the interval of zero length centered at 1. That is, the confidence that the marginal probability is equal to 1 is 100% and the variance of the sum of relative frequencies is 0. Regards, Maxim Makatchev Lotfi Zadeh wrote: > X and Y are random variables taking values in the set (1, 2, ..., > n). The entries in the joint probability matrix, P, are of the form > "approximately aij," where the aij take values in the unit interval > and add up to unity. What is the marginal probability distribution of > X? Two special cases: (a) "approximately aij," is interpreted as an > interval centering on aij; and (b) "approximately aij," is interpreted > as a fuzzy triangular number centering on aij. > > Warm regards to all, > > Lotfi > _______________________________________________ > uai mailing list > uai@ENGR.ORST.EDU > https://secure.engr.oregonstate.edu/mailman/listinfo/uai > _______________________________________________ uai mailing list uai@ENGR.ORST.EDU https://secure.engr.oregonstate.edu/mailman/listinfo/uai
