  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.
