Backward Filtering Forward Guiding

TL;DR

This paper introduces a versatile framework called Backward Filtering Forward Guiding (BFFG) for probabilistic inference in complex stochastic processes on DAGs, applicable to both discrete and continuous time, especially when transition densities are intractable.

Contribution

The paper presents a novel BFFG scheme that combines backward information filtering with forward guiding, enabling efficient inference in intractable stochastic models on DAGs.

Findings

BFFG produces weighted samples from the posterior distribution.

The method integrates well with MCMC and particle filtering.

Demonstrated on a branching diffusion process on a directed tree.

Abstract

We develop a general methodological framework for probabilistic inference in discrete- and continuous-time stochastic processes evolving on directed acyclic graphs (DAGs). The process is observed only at the leaf nodes, and the challenge is to infer its full latent trajectory: a smoothing problem that arises in fields such as phylogenetics, epidemiology, and signal processing. Our approach combines a backward information filtering step, which constructs likelihood-informed potentials from observations, with a forward guiding step, where a tractable process is simulated under a change of measure constructed from these potentials. This Backward Filtering Forward Guiding (BFFG) scheme yields weighted samples from the posterior distribution over latent paths and is amenable to integration with MCMC and particle filtering methods. We demonstrate that BFFG applies to both discrete- and continuous-time models, enabling probabilistic inference in settings where standard transition densities are intractable or unavailable. Our framework opens avenues for incorporating structured stochastic dynamics into probabilistic programming. We numerically illustrate our approach for a branching diffusion process on a directed tree.