A Gibbs sampler for a class of random convex polytopes

We present a Gibbs sampler for the Dempster--Shafer (DS) approach to statistical inference for Categorical distributions. The DS framework extends the Bayesian approach, allows in particular the use of partial prior information, and yields three-valued uncertainty assessments representing probabilities ``for'', ``against'', and ``don't know'' about formal assertions of interest. The proposed algorithm targets the distribution of a class of random convex polytopes which encapsulate the DS inference. The sampler relies on an equivalence between the iterative constraints of the vertex configuration and the non-negativity of cycles in a fully connected directed graph. Illustrations include the testing of independence in $2\times 2$ contingency tables and parameter estimation of the linkage model.