On the Geometry of Message Passing Algorithms for Gaussian Reciprocal Processes

TL;DR

This paper introduces belief propagation for Gaussian reciprocal processes and links its convergence to stability theory, addressing inference challenges in loopy graphical models.

Contribution

It develops belief propagation methods for Gaussian reciprocal processes and connects their convergence properties to stability analysis of positive systems.

Findings

Belief propagation can be applied to Gaussian reciprocal processes.

Convergence of belief propagation relates to stability of positive systems.

Provides theoretical insights into inference in loopy graphical models.

Abstract

Reciprocal processes are acausal generalizations of Markov processes introduced by Bernstein in 1932. In the literature, a significant amount of attention has been focused on developing dynamical models for reciprocal processes. Recently, probabilistic graphical models for reciprocal processes have been provided. This opens the way to the application of efficient inference algorithms in the machine learning literature to solve the smoothing problem for reciprocal processes. Such algorithms are known to converge if the underlying graph is a tree. This is not the case for a reciprocal process, whose associated graphical model is a single loop network. The contribution of this paper is twofold. First, we introduce belief propagation for Gaussian reciprocal processes. Second, we establish a link between convergence analysis of belief propagation for Gaussian reciprocal processes and stability theory for differentially positive systems.