  We show that any joint probability mass function (PMF) can be expressed as a
product of parity check factors and factors of degree one with the help of some
auxiliary variables, if the alphabet size is appropriate for defining a parity
check equation. In other words, marginalization of a joint PMF is equivalent to
a soft decoding task as long as a finite field can be constructed over the
alphabet of the PMF. In factor graph terminology this claim means that a factor
graph representing such a joint PMF always has an equivalent Tanner graph. We
provide a systematic method based on the Hilbert space of PMFs and orthogonal
projections for obtaining this factorization.
