A Hilbert space of probability mass functions and applications on the sum-product algorithm

In this paper a Hilbert space structure of probability mass functions (PMF) will be presented. The tools provided by the Hilbert space, specifically the norm and the inner product, may be useful while analyzing and improving the sum-product algorithm in many aspects. Our approach provides a metric distance between PMFs and a new point of a view of the log-likelihood ratio (LLR) such that the LLR representation is nothing but a Hilbert space representation.