stochasticity of sampling times but we condition on them as being fixed.

![[]{data-label="fig:hetTaj"}](markov_chain-eps-converted-to)

Let us introduce some notation. Let $m$ be the number of sampling time points and $n$ be the total number of samples. Let ${\textbf{n}}{}=(n_1,\ldots,n_{m})$ denote the number of sequences collected at times ${\textbf{s}}{}=(s_{1},\ldots,s_{m})$, with $s_{1}=0$ denoting the present time, and $s_{j}>s_{j-1}$ for $j=2,\ldots,m$ (time goes from present towards the past). We refer to the sequences counted in $n_i$ as “belonging to sampling group $s_i$”. Let ${\textbf{t}}{}=(t_{n+1}, \ldots, t_{2})$ be the vector of coalescent times with $t_{n+1}=0<t_{n}<...<t_{2}$; these are the times when two lineages have a common ancestor. Note that the subscript in $t_{k}$ does not indicate the current number of lineages, as it is often done in the coalescent literature, but it indicates the number of lineages that have yet to coalesce (some sequences may not have been sampled yet). We use the rank order of the coalescent events (bottom-up) to label the internal nodes of the genealogy. That is, the node corresponding to the coalescent event occurring at time $t_n$ is labeled $1$ (see $t_{10}$ in Figure \[fig:hetTaj\]), the node corresponding to the coalescence event occurring at time $t_{n-1}$ is labeled $2$, *etc*. We refer to 