Mesh Invariant Infinite Dimensional Adaptive MCMC for Latent Gaussian Processes
We introduce mesh-invariant adaptive Markov chain Monte Carlo methods for Gaussian-process posteriors arising in infinite-dimensional Bayesian inference. In function-space MCMC, posterior distributions are defined by a change of measure with respect to a Gaussian prior, making absolute continuity essential for valid proposal construction. Standard adaptive schemes that modify means or scales in the full discretized space can destroy this property, leading to proposal measures that become singular in the infinite-dimensional limit. To avoid this, we adapt only an active finite-dimensional subspace of the Gaussian-process representation while preserving the prior dynamics on inactive coordinates. This yields two adaptive proposals, pCNLV and pCNMV, which extend preconditioned Crank--Nicolson and Crank--Nicolson Langevin methods by learning posterior scale, and in pCNMV also posterior mean structure, on the data-informed subspace without introducing discretization-dependent Gaussian density ratios. The resulting samplers retain the mesh robustness of function-space methods while improving efficiency through local adaptation. Experiments on a Darcy-flow inverse problem and Bayesian logistic regression demonstrate consistent efficiency gains, including an approximately fourfold improvement in effective sampling efficiency for Darcy flow.