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Neural Backward Filtering Forward Guiding

NeurIPS 2026 · Probabilistic and scientific modeling › Dynamical systems · approach: State-space inference for stochastic systems · 0 citations (OpenAlex)

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Proposes NBFFG, a variational smoothing framework for nonlinear discrete and continuous processes on trees that corrects a closed-form linear-Gaussian proxy guide with neural residuals (Gaussian mixtures or controlled SDEs), trained with unbiased path-subsampled ELBO.

Problem

Smoothing (posterior inference of latent internal states) in nonlinear discrete and continuous stochastic processes on trees with sparse leaf observations is intractable because Doob's h-transform requires backward messages that have no closed form.

Approach

Build the variational posterior as a guided proposal from a cheap proxy linear-Gaussian backward filter (closed-form guide), then learn neural residual corrections (Gaussian-mixture transitions or controlled-SDE residual controls) shared across edges via path-context embeddings, with a Horvitz-Thompson-unbiased path-subsampled NELBO.

Key results

• On discrete LG benchmark 1A with the canonical (I,0) proxy, neural correction reduces the relative NELBO gap from 0.913 to 0.035 and average marginal KL from 1.861 to 0.118.
• On benchmark 2B early-commitment regime, normalized ESS improves from 0.273 (guide only) and 0.271 (guided SMC) to 0.529 for NBFFG.
• In the bimodal 2B regime, NBFFG is the only compared method that combines mode coverage with diverse paths, though its normalized ESS is low (0.0076).

Surprising findings

• Neural correction does not fully recover accuracy for some sign-flipped proxies in the discrete benchmark; marginal mean/covariance gains are negative for several proxies (e.g., covariance error worsens for (-A,b)).

Limitations

• Normalized ESS in the bimodal double-well regime remains very low (0.0076), so robust multimodal path-space correction is still challenging.
• Training still backpropagates through sampled root-to-leaf paths, and transfer across unseen topologies is not handled by the current amortization.
• Continuous-edge results are limited by fixed Euler-Maruyama discretization (50 steps per edge), so reference errors include discretization error.

Experimental setup

Datasets: synthetic Gaussian tree benchmarks synthetic OU tree benchmarks double-well diffusion synthetic benchmark butterfly wing outline landmark data

Benchmarks: Gaussian tree exact-posterior benchmarks (1A, 1B) nonlinear multimodal smoothing (2A) double-well diffusion path smoothing (2B) phylogenetic butterfly shape inference

Metrics: relative NELBO gap average marginal KL marginal mean error marginal covariance error normalized ESS

Baselines: No guidance ablation guided pCN guided SMC guide only Rauch-Tung-Striebel smoother (exact reference)

Builds on: backward filtering forward guiding Doob h-transform Girsanov theorem FiLM

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