lotfi zadeh gzrdaeecsberkeleyedu realworld probabilities imprecise reason move age machine intelligence mechanized decisionmaking problem deal imprecise probabilities certain grow visibility importance major contribution theory imprecise probabilities peter walleys 1991 book statistical reasoning imprecise probabilities london chapman hall since considerable progress made yet obvious problem computation imprecise probabilities intrinsically complex far definitive solution case point posted uai list september 22 2005 seemingly simple problem simple solution following broadened version problem presented simplest version problem perception even simple problem computationally nontrivial simple solution solutions problems b c problem 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 four special cases approximately aij interpreted interval centering aij b approximately aij interpreted triangular fuzzy number centering aij c approximately aij interpreted uniform probability distribution interval centering aij approximatelt aij interpreted triangular probability density function centering aij warm regards lotfi lotfi zadeh professor graduate school computer science division department electrical engineering computer sciences university california berkeley ca 94720 1776 director berkeley initiative soft computing bisc wed 06 aug 2008 112925 0800 uai imprecise probabilitiesa simple yet computationally nontrivial problem