On the Geometry of Message Passing Algorithms for Gaussian Reciprocal Processes
This work addresses inference challenges in acausal processes for researchers in machine learning and control theory, representing an incremental advancement by adapting existing methods to a specific graphical model.
The paper tackles the smoothing problem for Gaussian reciprocal processes by introducing belief propagation and linking its convergence analysis to stability theory for differentially positive systems, establishing a connection between these areas.
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.