Fig. 4

Numerical tests of our algorithms. The statistics \(t_{\text {MRS}}\), \(\sigma\), and q computed by exact algorithms were compared with those estimated using sampled histories of nucleotide substitutions. The y-axes represent the relative difference between the values from the exact algorithms and those obtained by approximate sampling algorithms. The x-axes show the dependency on the number of sampled histories and the number of discrete points in the phylogenetic tree from which the states were sampled