Automated generation of initial points for adaptive rejection sampling of log-concave distributions

Adaptive rejection sampling requires that users provide points that span the distribution’s mode. If these points are far from the mode, it significantly increases computational costs. This paper introduces a simple, automated approach for selecting initial points that uses numerical optimization to quickly bracket the mode. When an initial point is given that resides in a high-density area, the method often requires just four function evaluations to draw a sample—just one more than the sampler’s minimum. This feature makes it well-suited for Gibbs sampling, where the previous round’s draw can serve as the starting point.