Backward Filtering Forward Guiding

We study smoothing for discrete- and continuous-time stochastic processes on directed acyclic graphs (DAGs) when observations are available only at the leaf nodes, a problem common in phylogenetics, epidemiology, and signal processing. We introduce a unified framework built around guiding (also called twisting): a change-of-measure defined by guiding functions that transforms the original process into a guided process whose distribution approximates the smoothing distribution. The Radon-Nikodym derivative quantifies the discrepancy between the two measures. On directed trees, guiding functions are obtained via a backward-filtering step. By isolating backward filtering and forward guiding as elementary operations, we show that the approach extends beyond traditional state-space models and particle filters. We also generalize guiding to edges governed by continuous-time dynamics, using the change-of-measure construction described by Palmowski and Rolski (2002). The versatility of the framework is illustrated with two numerical examples: (i) a diffusion model for shape deformation on a tree, and (ii) inference in a factorial hidden Markov model.