Optimal damping function h(t) in the EnSF posterior score

Determine the mathematically optimal choice of the damping function h(t) used in the Ensemble Score Filter’s posterior score update s_{k|k}(x,t) = s_{k|k-1}(x,t) + h(t) · grad_x log p(y_k | x), subject to the constraints that h(t) is monotonically decreasing on the pseudo-time interval [0, T] with h(0) = 1 and h(T) = 0.

Background

In the score-based filtering formulation, the posterior score function combines the prior score with the likelihood score through a time-dependent damping function h(t): s_{k|k}(x,t) = s_{k|k-1}(x,t) + h(t) * grad_x log p(y_k | x). The function h(t) controls how observation information is incorporated along the diffusion model’s pseudo-time trajectory from T to 0.

The authors adopt h(t) = T − t for their SQG experiments and impose general constraints on h(t): it should be monotonically decreasing with h(0) = 1 and h(T) = 0. However, multiple choices for h(t) are possible, and identifying a mathematically optimal form is explicitly stated as an unresolved problem, with potential implications for accuracy and stability of EnSF’s posterior sampling.