Factorization of Joint Probability Mass Functions into Parity Check Interactions

TL;DR

This paper demonstrates that any joint probability mass function can be factorized into parity check interactions using auxiliary variables, linking probabilistic modeling with coding theory through a systematic Hilbert space approach.

Contribution

It introduces a novel factorization method for joint PMFs into parity check factors, bridging probabilistic models and coding theory with a systematic Hilbert space framework.

Findings

Any joint PMF can be expressed as parity check factors with auxiliary variables.

Factor graphs of joint PMFs have equivalent Tanner graph representations.

Provides a systematic Hilbert space-based method for factorization.

Abstract

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.