Title: Geometric Variational Inference: Elliptic Multi-Marginal Schrödinger Bridge, Anchor Compatibility, and Rates Entropic to Wasserstein
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
Keywords: Variational Inference, Schrödinger Bridge, Bayesian Smoothing, Smoothers, HJB–Fokker–Planck, Optimal Transport, Information Geometry, Multi-Marginal, IPFP/Sinkhorn, Ornstein–Uhlenbeck, Kalman–Bucy, Rauch–Tung–Striebel, Hellinger–Kantorovich (Reaction–Transport) Geometry

Abstract: We study variational smoothing as path-space inference in which time-marginals must remain compatible with a single evolution between observations. Our main result shows that path-space variational inference coincides with a multi-marginal Schrödinger bridge whose anchors are the posterior time-marginals, via the Gibbs Donsker Varadhan identity. This induces an Onsager-Fokker geometry: diffusion determines the metric tensor while drift enters through Fokker-Planck/HJB constraints that select the curve; in the linear-Gaussian case this recovers the Rauch-Tung-Striebel smoother. We further characterise limiting regimes as diffusion varies (convergence to $W_2$ displacement geodesics with segment-wise rates; large: convergence to mixture geodesics). Finally, we present a log-domain multi-marginal solver that computes posterior paths and provides theory-driven diagnostics on a controlled OU-GMM benchmark.

TL;DR: Prove that path-space VI is a multi-marginal Schrödinger bridge, inducing an Onsager–Fokker geometry, establish limiting regimes with rates and extend to unbalanced observations via VI–HK, and provide a stable IPFP