Concentration inequality for evolutionary trees

Maximum likelihood inferred topologies are commonly used to draw conclusions in evolutionary biology and molecular evolution. Considering the sampling error when estimating the topology is a critical issue. Bootstrap-based methods are the most popular tools to assess the robustness of clades, i.e. the stability of a tree and subtrees. Unfortunately, there is no analytical result to connect the bootstrap values to the sampling variability, or at least to the number of sites and species in the study. Using concentration measure tools, we first bound the variations of the computed likelihood around its true value and then bound the sampling variability of likelihood as measured by bootstrap. In particular and unlike most bootstrap-based methods, these bounds are explicitly sensitive to both the number of species and of nucleotides.