Synchronized step multilevel Markov chain Monte Carlo

TL;DR

SYNCE addresses the computational burden of Bayesian inverse problems solved via multilevel MCMC by introducing a synchronized-step, common random-number coupling that preserves strong cross-level correlation even when posteriors differ. The authors prove a unique invariant measure and geometric ergodicity for the SYNCE kernel, derive an explicit convergence-rate bound, and show that adaptation and resynchronization further boost performance. Through experiments on shifting/rotating Gaussians, a prey-predator model, and groundwater flow, SYNCE yields substantial variance reduction and scalable efficiency, outperforming existing couplings in non-overlapping posterior regimes. The work offers a robust framework for ML-MCMC coupling with potential extensions to transport maps and unbiased estimators, enabling efficient Bayesian inference on large fidelity hierarchies.

Abstract

We propose SYNCE (synchronized step correlation enhancement), a new algorithm for coupling Markov chains within multilevel Markov chain Monte Carlo (ML-MCMC) estimators. We apply this algorithm to solve Bayesian inverse problems using multiple model fidelities. SYNCE is inspired by the concept of common random number coupling in Markov chain Monte Carlo sampling. Unlike state-of-the-art methods that rely on the overlap of level-wise posteriors, our approach enables effective coupling even when posteriors differ substantially. This overlap-independence generates significantly higher correlation between samples at different fidelity levels, improving variance reduction and computational efficiency in the ML-MCMC estimator. We prove that SYNCE admits a unique invariant probability measure and demonstrate that the coupled chains converge to this measure faster than existing overlap-dependent methods, particularly when models are dissimilar. Numerical experiments validate that SYNCE consistently outperforms current coupling strategies in terms of computational efficiency and scalability across varying model fidelities and problem dimensions.

Figures (9)

• Figure 1: Comparison of the four coupling methods for two posteriors at levels $\ell-1$ and $\ell$ that are different and do not overlap.
• Figure 2: Posterior distributions for the Gaussian examples at different levels. The figure on the left corresponds to \ref{['subsec:shifting-gaussian']} and the figure on the right corresponds to \ref{['subsec:rotating-shifting-gaussian']}.
• Figure 3: Scatter plots and histograms of samples from the different coupling algorithms for the shifting Gaussian example. For each level, the X-axis represents the samples at the $\ell$'th level and the Y-axis represents the samples at the $(\ell-1)$'th level. The red line on the histogram plots represents the true posterior distribution at the respective levels. Existing methods either fail to correctly sample from the marginals (row 1), produce diffused scatter plots with weak correlation despite correct marginals (row 2), or exhibit degrading diagonal concentration as posteriors diverge (row 3). SYNCE (row 4) consistently produces tightly clustered points along a dominant diagonal across all levels -- indicating robust correlation. The adaptive diagonal slope is not constrained to one, reflecting the relative shift between the posteriors while maintaining right marginals.
• Figure 4: Pearson correlation coefficient between samples of all levels for the shifting Gaussian example. The coefficient is a linear measure of how well samples are correlated. Higher correlations yield greater variance reduction and we see that the SYNCE coupling algorithm achieves the greatest correlation, particularly for the coarser levels of the hierarchy.
• Figure 5: Comparison of the maximal coupling method and baseline SYNCE on the rotating-shifting Gaussian example (first dimension). (a) Level-wise correlation showing SYNCE's advantage at lower levels but disadvantage at finer levels. (b) Autocorrelation at the finest level, confirming comparable sampling efficiency because of similar proposal distributions.

Theorems & Definitions (6)

• Lemma 4.1
• Theorem 4.1: Geometric ergodicity of the SYNCE coupled kernel
• Lemma A.1: $\psi-$irreducibility
• Lemma A.2: Minorization condition
• Lemma A.3: Drift condition
• Theorem B.1: $\epsilon$-cost of ML-MCMC with SYNCE coupling