Posterior Convergence without Force Convergence: Resolution-Stable Sampling for Rough Bayesian Inverse Problems

Abstract

Bayesian targets may converge under model refinement even when the exact sensitivities used by gradient-based samplers do not. We study this probability--sensitivity mismatch and its consequences for Metropolized Hamiltonian proposals. A vanishing-amplitude wiggly-energy model first gives the basic analytic obstruction: the potential perturbation tends to zero while its classical derivative is of order $r_\varepsilon/\varepsilon$. We then show that the same scaling arises naturally in a periodic elliptic inverse problem, where homogenization makes the forward map and Gaussian likelihood converge while differentiation with respect to a microscopic scale parameter retains an $O(1)$ oscillatory contribution. This provides a PDE origin for the single-scale wiggly mechanism. The main construction concerns a more demanding nested Weierstrass hierarchy, interpreted as an analytically tractable prototype for repeated corrector contributions across geometrically separated scales. There all previously resolved scales persist, adjacent classical-force increments grow geometrically like $(ab)^N$, and the limiting rough component may fail to possess a classical derivative. In this self-similar setting the matched Jackson quotient is structurally adapted to the refinement through dilation covariance and exact finite closure. Uniform negative-log-likelihood approximation yields explicit total-variation, Hellinger, and bounded quantity-of-interest bounds. Measurable kick--drift--kick maps remain exact after Metropolis correction, local field convergence propagates to fixed-length proposals and kernels, and the first classical HMC half-kick can have no fixed-step refinement limit. Numerical experiments on scale-structured inverse problems test the resulting resolution-stability mechanism across one- and two-dimensional inverse problems.