Facilitating the Gibbs Sampler: The Gibbs Stopper and the Griddy-Gibbs Sampler

Abstract The article briefly reviews the history, literature, and form of the Gibbs sampler. An importance sampling device is proposed for converting the output of the Gibbs sampler to a sample from the exact posterior. This Gibbs stopper technique is also useful for assessing convergence of the Gibbs sampler for moderate sized problems. Also presented is an approach for implementing the Gibbs sampler in nonconjugate situations. The basic idea is to approximate the true cdf of each conditional distribution by a piecewise linear function and then sample from the approximation. Questions relating to the number of nodes in the approximation, gap size between successive nodes, and the treatment of unbounded intervals for a given conditional are discussed. The methodology is illustrated using a genetic linkage model, a nonlinear regression model, and the Cox model.