Auto-validating von Neumann Rejection Sampling from Small Phylogenetic Tree Spaces

In phylogenetic inference one is interested in obtaining samples from the posterior distribution over the tree space on the basis of some observed DNA sequence data. The challenge is to obtain samples from this target distribution without any knowledge of the normalizing constant. One of the simplest sampling methods is the rejection sampler due to von Neumann. Here we introduce an auto-validating version of the rejection sampler, via interval analysis, to rigorously draw samples from posterior distributions, based on homologous primate mitochondrial DNA, over small phylogenetic tree spaces.