Onsager-Machlup Posterior Transport for Deep Gaussian Processes

Approximate inference over inducing variables is the central computational bottleneck of Deep Gaussian Processes (DGPs). Existing methods either fit an explicit density qφ(bU) by an ELBO (DSVI, IPVI, DDVI, DBVI) or sample by MCMC (SGHMC). We instead frame DGP inference as posterior transport: learn a deterministic sampler that maps a tractable reference measure to posterior-relevant inducing variables, regularised by a path prior derived from the Doob-bridged reference diffusion. Our realisation, OM-Path (formally FBVI-bridge-Path), uses Song's probability-flow ODE applied to DBVI's Doob-bridged forward SDE; the reference drift is closed-form from the bridge marginal coefficients (no score matching) and the path regulariser is the Onsager--Machlup action. At the finite-ε value used at training, the objective is the negative log unnormalised density of a tempered Doob-bridge path posterior, and Theorem 1 identifies it with the same posterior's small-noise MAP path via the Freidlin--Wentzell LDP. Two strict path-space ELBO variants on the same bridge backbone (FFJORD log-det; OM-regularised CNF) are derived as ablations. Under a matched-seed paired Wilcoxon test against DBVI on seven UCI regression benchmarks, OM-Path delivers statistically significant wins on the two largest datasets (textitpower: p!=!0.014, NLL 0.012 matching the DSVI baseline of 0.017; textitprotein: p!=!0.002, RMSE 0.716 vs. 0.764, NLL 1.086 vs. 1.149), statistical ties on textityacht / textitqsar, and concedes textitboston / textitenergy / textitconcrete to DBVI on small-N noisy data. The strict-ELBO variants do not clear DBVI on any UCI metric: in this regime, reducing the variance of the path objective dominates exact-density tracking.