Contour Monte Carlo: Sampling via Energy Level Sets

Efficiently sampling from complex distributions is central to Bayesian inference, yet even modern MCMC samplers can struggle to converge on geometrically challenging targets. In this work, we identify a decomposition of sampling error into mismatch in the potential energy (PE) marginal and average conditional mismatch across PE level sets. This motivates *Contour Monte Carlo* (CMC), a two-stage framework which corrects each of these components by first drawing energies from the PE marginal and then sampling from the corresponding level sets. For a large class of radial targets, including Gaussian and Student's $t$ distributions, both stages can be done exactly. For general targets, CMC approximates the PE marginal using umbrella sampling with MBAR and then runs constrained MCMC on the sampled level sets, initialised from energy-matched umbrella samples. We show that this procedure essentially reduces the global sampling problem to estimating the PE marginal (a one-dimensional quantity), after which the downstream constrained sampling usually converges quickly. Furthermore, the level-set chains are embarrassingly parallel and well-suited to modern accelerator hardware. We test CMC on a diverse suite of benchmarks and find that, on the most difficult of these, it delivers decisively lower sampling error than HMC and NUTS baselines at a fixed computational budget, while staying competitive on simpler targets.