at $z$ polymorphic (mutating) sites at a non-recombining contiguous segment of DNA of organisms with low mutation rate. Under these assumptions, a widely studied mutation model is the *infinite sites model* (ISM) [@kim69; @wat75] with Poissonian mutation, which corresponds to throwing a Poisson point process of mutations on the branches of [**g**]{} at rate $\mu$ such that every mutation occurs at a different site and no mutations are hidden by a second mutation affecting the same site.

An important consequence of the ISM is that [**Y**]{} can be represented as an incidence matrix ${\textbf{Y}}{}_1$ and a frequency counts matrix ${\textbf{Y}}{}_2$. ${\textbf{Y}}{}_1$ is a $k \times z$ matrix with $0$-$1$ entries, where $0$ indicates the ancestral type and $1$ indicates the mutant type; $k$ is the number of unique sequences (or haplotypes) observed in the sample, and the columns correspond to polymorphic sites. ${\textbf{Y}}{}_2$ is a $k \times m$ count matrix where the $(i,j)$th entry denotes how many haplotype $i$ sequences belonging to group $s_j$ are sampled. For example, the $n = 10$ sequences defined by the realizations of the ancestral and mutation processes depicted in Figure \[fig:coalescent\] can be summarized into ${\textbf{Y}}_1$ and ${\textbf{Y}}_2$ displayed in Figure \[perfect\_phylo\](A).

![[]{data-label="perfect_phylo"}](perf_phylo2-eps-converted-to){width="100.00000%"}

${\textbf{Y}}{}_1$ 