Bayesian Sampling with Approximate Transport Geometry via Residual-Slice Correction

Approximate transport maps can facilitate exploration of a Bayesian target distribution, but the resulting samples generally do not follow that distribution. To address this problem, we develop residual-slice correction, a sampling framework that combines slice sampling with an approximate transport map held fixed during sampling. Each iteration uses a slice variable to represent the residual left by the map and updates the state while preserving the conditional distribution on the resulting feasible set. To assess sampling efficiency, we derive a lower bound on the corrected chain's Dirichlet-form gap using a reference Markov kernel. The bound separates movement within each feasible set, the transport-reference comparison, and reference mixing, while projected diagnostics examine the first two factors. Numerical experiments show that residual-slice correction recovers summaries and shape diagnostics distorted by approximate transport; they also show that the choice of Markov update within each feasible set substantially affects mixing efficiency, and that the corrected chains have lower serial dependence after normalizing-flow training. Overall, the framework retains the geometric benefits of approximate transport while preserving the target distribution.