Transport-map proposals for efficient Markov Chain Monte Carlo

SUMMARY Efficient Markov chain Monte Carlo (MCMC) sampling from posterior distributions remains a central challenge in Bayesian geophysical inversion. Recent developments in computational statistics and optimal transport suggest that MCMC efficiency can be improved by reparametrizing the sampling problem—specifically, by learning an invertible mapping that recasts the target distribution onto a simpler reference distribution. Here, we introduce a Metropolis–Hastings framework that leverages transport maps parametrized by invertible neural networks. These maps are trained on preliminary MCMC samples from the target distribution and used to propose new samples in a fixed reference space, where proposal design is independent of the target’s structure. The proposed samples are transformed back to the target space via the inverse map, and accepted or rejected according to a modified Metropolis–Hastings criterion. As sampling proceeds, the transport maps are updated, yielding proposals increasingly well adapted to the shape of the target distribution. Across a suite of numerical tests—including a 2-D Rosenbrock distribution, a 3-D earthquake location problem and Gaussian mixtures up to 16 dimensions—transport-map-driven samplers consistently outperform standard MCMC, reducing integrated autocorrelation times by factors of 2.5 to over 6 (or equivalently, yielding sample sets 2.5–6 times larger for the same number of forward evaluations). This improvement comes at the non-negligible cost of training one or more transport maps, which we quantify systematically. We also provide a quantitative criterion for weighing training cost against sampling speed-up. This shows that transport-map MCMC is advantageous whenever the forward problem is non-trivial, making it a promising approach for Bayesian sampling in geophysics and beyond.