The Hilbert Space of Probability Mass Functions and Applications on Probabilistic Inference

TL;DR

This thesis introduces a Hilbert space framework for probability mass functions, enabling a novel factorization method that reveals algebraic and independence relations, leading to new probabilistic inference algorithms and hardware implementations.

Contribution

It proposes a new Hilbert space-based factorization of multivariate pmfs that uncovers variable relations and enables the use of channel decoders for inference tasks.

Findings

The factorization reveals algebraic relations between variables.

It determines conditional independence relations.

A MIMO detection algorithm with reduced complexity was developed.

Abstract

The Hilbert space of probability mass functions (pmf) is introduced in this thesis. A factorization method for multivariate pmfs is proposed by using the tools provided by the Hilbert space of pmfs. The resulting factorization is special for two reasons. First, it reveals the algebraic relations between the involved random variables. Second, it determines the conditional independence relations between the random variables. Due to the first property of the resulting factorization, it can be shown that channel decoders can be employed in the solution of probabilistic inference problems other than decoding. This approach might lead to new probabilistic inference algorithms and new hardware options for the implementation of these algorithms. An example of new inference algorithms inspired by the idea of using channel decoder for other inference tasks is a multiple-input multiple-output (MIMO) detection algorithm which has a complexity of the square-root of the optimum MIMO detection algorithm.