The hit-and-run sampler: A globally reaching Markov chain sampler for generating arbitrary multivariate distributions

The problem of efficiently generating general multivariate densities via a Monte Carlo procedure has experienced dramatic progress in recent years through the device of a Markov chain sampler. This procedure produces a sequence of random deviates corresponding to a random walk over the support of the target distribution. Under certain regularity conditions, the corresponding Markov chain converges in distribution to the target distribution. Thus the sample of points so generated can serve as a statistical sample of points drawn from the target distribution. A random walk that can globally reach across the support of the distribution in one step is called a Hit-and-Run sampler. Hit-and-Run Markov chain samplers offer the promise of faster convergence to the target distribution than conventional small step random walks. Applications to optimization are considered.