Variable Elimination in Fourier Domain

Probabilistic inference is a key computational challenge in statistical machine learning and artificial intelligence. The ability to represent complex high dimensional probability distributions in a compact form is the most important insight in the field of graphical models. In this paper, we explore a novel way to exploit compact representations of high-dimensional probability distributions in approximate probabilistic inference algorithms. Our approach is based on discrete Fourier Representation of weighted Boolean Functions, complementing the classical method to exploit conditional independence between the variables. We show that a large class of probabilistic graphical models have a compact Fourier representation. This theoretical result opens up an entirely new way of approximating a probability distribution. We demonstrate the significance of this approach by applying it to the variable elimination algorithm and comparing the results with the bucket representation and other approximate inference algorithms, obtaining very encouraging results.