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 $q_{\varphi }(U)$ 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), applies Song's probability-flow ODE 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 $ϵ$ used in training, the objective is a data-fit term plus the classical Onsager–Machlup small-tube path prior; Theorem 1 identifies its small-noise limit with the conditional MAP path of the corresponding tempered Doob-bridge path posterior. Two strict path-space ELBO variants on the same backbone are derived as ablations. Under a matched-seed paired Wilcoxon test against DBVI on seven UCI regression benchmarks, OM-Path wins significantly on the two largest datasets (power $p=0.007$, NLL $0.009$ against the DSVI baseline of $0.017$; protein $p=0.002$, RMSE $0.718$ vs. $0.761$), ties on yacht / qsar, and concedes the small-$N$ noisy cells to DBVI. It also outperforms the strict-ELBO variants on $13/14$ UCI cells, and our gradient-variance measurements show a substantial Hutchinson penalty on power, though the size of that effect is dataset-dependent.