General Hit-and-Run Monte Carlo sampling for evaluating multidimensional integrals

We elaborate on the Hit-and-Run sampler, a Monte Carlo approach that estimates the value of a high-dimensional integral with integrand h((x) under bar)f((x) under bar) by sampling from a time-reversible Markov chain over the support of the density f. The Markov chain transitions are defined by choosing a random direction and then moving to a new point (x) under bar whose likelihood depends on f in that direction. The serially dependent observations of h(<(x)under bar (i)>) are averaged to estimate the integral. The sampler applies directly to f being a nonnegative function with finite integral. We generalize the convergence results of Belisle et al. [3] to unbounded regions and to unbounded integrands. Here convergence is of the point estimator to the value of the integral; this convergence is based on convergence in distribution of realizations to their limiting distribution f. An important application is determining properties of Bayesan posterior distributions. Here f is proportional to the posterior density and h is chosen to indicate the property being estimated. Typical properties include means, variances, correlations, probabilities of regions, and predictive densities.