Convergence of Gibbs Sampling: Coordinate Hit-and-Run Mixes Fast

Gibbs sampling, also known as Coordinate Hit-and-Run (CHAR), is a Markov chain Monte Carlo algorithm for sampling from high-dimensional distributions. In each step, the selects random coordinate and re-samples that distribution induced by fixing all other coordinates. While this has become widely used over past half-century, guarantees of efficient convergence have been elusive. We show convex body K in $${\mathbb {R}}^n$$ mixes $$O^{*}(n^9 R^2/r^2)$$ steps, where contains ball radius r R average distance point its centroid. give an upper bound on conductance Hit-and-Run, showing it strictly worse than or Ball Walk worst case.