Belief Propagation for Continuous State Spaces: Stochastic Message-Passing with Quantitative Guarantees

TL;DR

This paper introduces SOSMP, a stochastic message-passing method for continuous graphical models that guarantees convergence to the belief propagation fixed point using orthogonal series and Monte Carlo approximations.

Contribution

The paper presents SOSMP, a novel stochastic message-passing algorithm for continuous variables with proven convergence and practical guidelines for basis coefficient selection.

Findings

Converges to BP fixed point in tree-structured graphs.

Applicable to graphs with cycles under contractivity conditions.

Validated through simulations and optical flow application.

Abstract

The sum-product or belief propagation (BP) algorithm is a widely used message-passing technique for computing approximate marginals in graphical models. We introduce a new technique, called stochastic orthogonal series message-passing (SOSMP), for computing the BP fixed point in models with continuous random variables. It is based on a deterministic approximation of the messages via orthogonal series expansion, and a stochastic approximation via Monte Carlo estimates of the integral updates of the basis coefficients. We prove that the SOSMP iterates converge to a \delta-neighborhood of the unique BP fixed point for any tree-structured graph, and for any graphs with cycles in which the BP updates satisfy a contractivity condition. In addition, we demonstrate how to choose the number of basis coefficients as a function of the desired approximation accuracy \delta and smoothness of the compatibility functions. We illustrate our theory with both simulated examples and in application to optical flow estimation.