We propose a novel method for simulating conditioned diffusion processes (diffusion bridges) in Euclidean spaces. By training a neural network to approximate bridge dynamics, our approach eliminates the need for computationally intensive Markov Chain Monte Carlo (MCMC) methods or reverse-process modeling. Compared to existing methods, it offers greater robustness across various diffusion specifications and conditioning scenarios. This applies in particular to rare events and multimodal distributions, which pose challenges for score-learning- and MCMC-based approaches. We propose a flexible variational family for approximating the diffusion bridge path measure which is partially specified by a neural network. Once trained, it enables efficient independent sampling at a cost comparable to sampling the unconditioned (forward) process.

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.

Stochastically evolving geometric systems are studied in shape analysis and computational anatomy for modeling random evolutions of human organ shapes. The notion of geodesic paths between shapes is central to shape analysis and has a natural generalization as diffusion bridges in a stochastic setting. Simulation of such bridges is key to solving inference and registration problems in shape analysis. We demonstrate how to apply state-of-the-art diffusion bridge simulation methods to recently introduced stochastic shape deformation models, thereby substantially expanding the applicability of such models. We exemplify these methods by estimating template shapes from observed shape configurations while simultaneously learning model parameters.