Contour Monte Carlo: Sampling via Energy Level Sets

Efficient sampling from complex probability distributions is an important task across Bayesian inference and certain classes of deep generative models. Markov chain Monte Carlo (MCMC) remains one of the most widely used tools for this. However, even state-of-the-art samplers, such as the NUTS implementations in Stan and Turing.jl, can converge slowly on ill-conditioned, heavy-tailed, or near-degenerate targets. We identify that sampling error can be decomposed into two terms: mismatch in the potential energy (PE) marginal, and average conditional mismatch across PE level sets. We propose *Contour Monte Carlo (CMC)*, which corrects each of these in turn: first by drawing energies $u_i$ from the PE marginal, and then by sampling states on the corresponding level sets. For a broad family of radial targets, including the Gaussian and Student's $t$ distributions, both stages admit closed-form draws, so CMC is exact and requires no Markov chain. Beyond this, for general targets, we estimate the PE marginal using umbrella sampling with MBAR, then run constrained MCMC on the sampled level sets, initialised from energy-matched umbrella samples (which tend to lie in regions of high conditional mass). We show that this procedure essentially reduces the global sampling problem to approximating the one-dimensional PE marginal, after which posterior draws can be generated readily on demand. Furthermore, the level-set chains are embarrassingly parallel, and therefore well-suited to modern accelerator hardware. On a suite of challenging benchmarks -- composed of `posteriordb` models, synthetic targets, and real-world inference problems -- CMC converges substantially faster than HMC and NUTS baselines at a fixed computational budget.