Hit-and-run for numerical integration
Provides theoretical guarantees for hit-and-run integration, advancing tractability analysis for log-concave densities in high dimensions.
The paper studies numerical integration of bounded functions under log-concave densities on convex bodies, showing that the hit-and-run algorithm achieves refined tractability with error bounds for multi-run MCMC.
We study the numerical computation of an expectation of a bounded function with respect to a measure given by a non-normalized density on a convex body. We assume that the density is log-concave, satisfies a variability condition and is not too narrow. We consider general convex bodies or even the whole $\R^d$ and show that the integration problem satisfies a refined form of tractability. The main tools are the hit-and-run algorithm and an error bound of a multi run Markov chain Monte Carlo method.