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Mesh Invaiant Infinite Dimensional Adaptive MCMC for Latent Gaussian Processes

NeurIPS 2026 · Probabilistic and scientific modeling › Uncertainty and Gaussian processes · approach: Simulation-based inference

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Proposes mesh-invariant adaptive function-space MCMC (pCNL_V and pCN_MV) that adapts only a finite active subspace of the Gaussian prior's Karhunen–Loève modes, preserving equivalence to the prior, and reports efficiency gains on a Darcy flow inverse problem and Bayesian logistic regression.

Problem

Function-space MCMC for latent Gaussian-process posteriors deteriorates in efficiency when posterior mean/scale differ from the prior, and naive adaptation in the full discretized space can destroy absolute continuity with respect to the prior in the infinite-dimensional limit.

Approach

Adapt the Gaussian reference measure (covariance multipliers and mean) via KL-matching and diagonal inverse-Fisher scaling, but only on a truncated, growing set of leading prior KL modes, so the adapted reference stays Feldman–Hajek equivalent to the prior at every iteration.

Key results

• On Darcy flow, pCN_MV achieves ESS 7742.5 versus 1747.6 for adaptive pCN over 48000 post-burn-in samples (more than four-fold).
• On logistic regression, pCNLV-F and pCNLV reach ESS/s of 59.3 and 51.5 on German versus 1.19 for pCN, and pCN_MV reaches 502 ESS/s on Pima versus 18.6 for pCN.
• On the Gaussian benchmark, truncated online pCN_MV reaches orders-of-magnitude larger ESS/s than pCN, while global and random-scan variants improve less.

Surprising findings

• Global (untruncated) adaptation converges much more slowly and remains noisy in leading coordinates, so adapting fewer modes works better than adapting all.

Limitations

• Adaptation is diagonal in the prior KL basis, so it cannot learn correlations between prior directions and requires a known prior basis and mode ordering.
• Darcy comparison only covers pCN vs pCN_MV; gradient-based variants were not compared due to expensive forward solves.
• Equivalence of the non-Gaussian target's optimal variance multipliers relies on an assumed variance regularity condition.

Experimental setup

Datasets: Australian Heart German Pima Ripley synthetic Darcy flow data

Benchmarks: Gaussian conjugate benchmark Darcy flow inverse problem Australian/Heart/German/Pima/Ripley logistic regression

Metrics: effective sample size ESS per second acceptance rate

Baselines: pCN pCNL AuxMarg adaptive pCN

Builds on: preconditioned Crank-Nicolson (pCN) preconditioned Crank-Nicolson Langevin (pCNL) Feldman-Hajek theorem DNA prior construction

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