An Auto-validating Rejection Sampler for Differentiable Arithmetical Expressions: Posterior Sampling of Phylogenetic Quartets

Abstract

Abstract. We introduce an efficient extension of a recently introduced auto-validating rejection sampler that is capable of producing independent and identically distributed (IID) samples from a large class of target densities with locally Lipschitz arithmetical expressions. Our extension is restricted to target densities that are differentiable. We use the centered form, as opposed to the natural interval extension, to get tighter range enclosures of the differentiable multivariate target density using interval extended gradient differentiation arithmetic. By using the centered form we are able to sample one hundred times faster from the posterior density over the space of phylogenetic trees with four leaves (quartets).