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

What the math gives to ML

The paper isolates a failure mode for refinement-based samplers: rough potentials can have stable posteriors while their classical derivatives become increasingly resolution-dependent. Its most transferable construction is a dilation-matched finite-difference field derived from the exact scale recursion of Weierstrass potentials. This field can be inserted into kick-drift-kick proposals and corrected by Metropolis acceptance without requiring it to be the gradient of a differentiable potential. The resulting sampler is testable on neural energy models with multiscale or nonsmooth components by measuring whether acceptance and effective sample size remain stable as resolution increases.