Backward Filtering Forward Guiding

TL;DR

This paper develops Backward Filtering Forward Guiding (BFFG), a general methodology for Bayesian smoothing of latent trajectories on directed acyclic graphs with leaf-only observations, applicable to both discrete- and continuous-time dynamics. It combines a backward information filter that computes likelihood-informed potentials with a forward guided process obtained via an exponential change of measure, enabling tractable sampling and unbiased posterior estimation of latent paths. The framework is designed to be compatible with MCMC and particle-filtering methods and extensible to general DAGs, with demonstrations across discrete kernels, continuous-time dynamics, and branching structures such as trees. The numerical illustrations include diffusion processes on trees and parameter-estimation experiments, highlighting BFFG’s potential for probabilistic programming and complex stochastic dynamics in domains like phylogenetics and shape analysis.

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.

Figures (8)

• Figure 1: Example of a tree with known root vertex $r$, with observations at vertices $5$, $6$ and $7$. A continuous time stochastic process evolves on the branches $(0, 3)$, $(3, 4)$, $(0, 1)$ and $(1, 2)$ which are coloured blue.
• Figure 2: A DAG with two leaves and two roots.
• Figure 3: Forward simulated paths from \ref{['eq:tanh_sde']} with $(\theta_0, \theta_1, \sigma_0, \sigma_1)=(0.0,0.65,0.1,0.4)$ on a 5 level tree with 121 nodes of which 81 are leaf nodes. Only the values at the leaf nodes are observed.
• Figure 4: Traceplots for the parameters $\theta_0,\theta_1,\sigma_0,\sigma_1$. Traceplots for $\theta_0$ and $\sigma_0$ are in blue; traceplots for $\theta_1$ and $\sigma_1$ are in orange. Green horizontal lines indicate true valeus; red horizontal lines show posterior mean after removing burnin samples.
• Figure 5: Densities after removing the first 2000 iterations which are considered burnin samples.

Theorems & Definitions (38)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Example 3.4
• Remark 1
• Definition 4.1
• Theorem 4.2
• proof
• Remark 2
• Definition 4.3