Monte Carlo on manifolds: sampling densities and integrating functions
For researchers needing to sample or integrate over constrained manifolds, this work provides a practical method that avoids second derivatives, but the novelty is incremental as it adapts existing projection-based MCMC to inequality constraints.
The paper presents an MCMC sampler for probability distributions on manifolds defined by equality and inequality constraints, using only first derivatives, and a multi-stage algorithm for integrating functions over such manifolds with single-run error estimates. The method is applied to compute entropies of sticky hard sphere systems, predicting transition temperatures between loops and chains.
We describe and analyze some Monte Carlo methods for manifolds in Euclidean space defined by equality and inequality constraints. First, we give an MCMC sampler for probability distributions defined by un-normalized densities on such manifolds. The sampler uses a specific orthogonal projection to the surface that requires only information about the tangent space to the manifold, obtainable from first derivatives of the constraint functions, hence avoiding the need for curvature information or second derivatives. Second, we use the sampler to develop a multi-stage algorithm to compute integrals over such manifolds. We provide single-run error estimates that avoid the need for multiple independent runs. Computational experiments on various test problems show that the algorithms and error estimates work in practice. The method is applied to compute the entropies of different sticky hard sphere systems. These predict the temperature or interaction energy at which loops of hard sticky spheres become preferable to chains.