Representing Belief Functions as Random Variables

Random sets are the obstacle for implementing belief functions in knowledge-based systems. The challenges include inefficient manipulations of subsets and the cognitive complexity of problem representations. The back-end knowledge management is yet another bottleneck in practice. Here, I propose representing subsets as integers, set operations as integer ones, and belief functions as functions of random integers, to meet these challenges. Using random variables, Dempster's rule of combination is reduced into matrix multiplications and its complexity is minimized. Fast Mobius transformation (FMT) is also dramatically improved. For example, to compute beliefs from a mass function in the power set with frame size n = 32, a Mobius inversion that takes 1057 years to accomplish using nonoptimized set operations or 15.8 years via the best existing FMT algorithm will take only one second via the random-variable based FMT. In addition, the new FMT forgoes the need to maintain and lookup any graphical structures and allows the application of FMT to any list of subsets, rather than the power set.