jason palmer mrktgulrucsdedu dear prof zadeh others view notion imprecise probabilities welldefined seems imagining something like experiment performed probability distribution drawn randomly set distributions experiment performed based distribution seems intermediary probabilities arent really probabilities model parameters still final distribution values outcome experiment model found marginalizing random probabilities essence probability event occurs according fixed distribution probability distribution cant apply one experiment whatever event observed either occur stationary distribution distribution impossible distinguish particular generating sequence random variables model yields final distribution marginalized possible might able talk distribution evolving time regarding probability questions c dont think specify marginal distributions probabilities aij advance might joint distribution satisfies marginals eg x takes values 123 probabilities p1 p2 p3 cant interval 05 example would possible find joint distribution results restricting probabilities certain admissible intervals uniform case joint density would occupy intersection hypercube simplex expectations would calculated integrating polytope normalizing volume weighted integration could performed triangular case think intersection linear varieties hypercubes called zonotopes may formulae volumes kind regards jason original message nhipljlbixengrorstedu mailtonhipljlbixengrorsteduon behalf lotfi zadeh sent wednesday november 09 2005 515 pm faqengrorstedu cc gzrdaeecsberkeleyedu subject uai imprecise probabilitiesa simple yet computationally nontrivial problem 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 114351 0800 uai imprecise probabilitiesa simple yet computationally nontrivial problem