Monte Carlo on manifolds: sampling densities and integrating functions

摘要

We describe and analyze some Monte Carlo methods for manifolds in Euclideanspace defined by equality and inequality constraints. First, we give an MCMCsampler for probability distributions defined by un-normalized densities onsuch manifolds. The sampler uses a specific orthogonal projection to thesurface that requires only information about the tangent space to the manifold,obtainable from first derivatives of the constraint functions, hence avoidingthe need for curvature information or second derivatives. Second, we use thesampler to develop a multi-stage algorithm to compute integrals over suchmanifolds. We provide single-run error estimates that avoid the need formultiple independent runs. Computational experiments on various test problemsshow that the algorithms and error estimates work in practice. The method isapplied to compute the entropies of different sticky hard sphere systems. Thesepredict the temperature or interaction energy at which loops of hard stickyspheres become preferable to chains.