Brownian Smoothing via the Driving Noise

Amortized posterior sampling offers a promising route to accurate and efficient smoothing. Learned smoothers can model complex trajectory posteriors, while drawing a sample still often requires full-trajectory modeling or a large population of particles. In this paper, we introduce Brownian Smoothing, which generates complete posterior trajectories in one rollout by conditioning the initial state and shifting the simulator's Gaussian driving noise. To maintain accuracy under one-rollout sampling, Brownian Smoothing is trained by conditional maximum likelihood on model simulations. And we prove that this objective is a forward-KL projection and that the optimal drift correction converges to the one induced by Doob's $h$-transform under grid refinement. Empirically, Brownian Smoothing closely matches exact posterior statistics and achieves the best marginal and joint scores on nonlinear, high-dimensional benchmarks. It also substantially reduces sampling time and memory compared with global flow and score-based baselines.