which reads: for $k=3,...,n+1$, $$\label{prior_time}
p(t_{k-1}\mid t_{k},\mathbf{s},\mathbf{n},(N_{e}(t))_{t\geq 0})=\frac{C_{0,k-1}}{N_{e}(t_{k-1})} \exp \left\lbrace - \int_{I_{0,k-1}} \frac{C_{0,k-1}}{N_{e}(t)}{{\rm d}}t+\sum^{m}_{i=1}\int_{I_{i,k-1}} \frac{C_{i,k-1}}{N_{e}(t)}{{\rm d}}t\right\rbrace,$$ where $t_{n+1}=0$ by convention, $C_{i,k}:=\binom{n_{i,k}}{2}$, and the integral over $I_{i,k-1}$ is zero if there are less than $i$ sampling times between $t_k$ and $t_{k-1}$. The distribution of the holding times defined above corresponds to the same distribution of holding times in the heterochronous Kingman coalescent [@rod99]. Although the heterochronous Tajima coalescent takes value on a different state space, it remains true that every pair of extant lineages coalesces at equal rate. Finally, given [**n**]{}, [**s**]{} and ${\textbf{t}}{}$, a complete realization of the Tajima heterochronous $n$-coalescent chain can be uniquely identified with a partially labeled binary ranked tree shape $g$ of $\mathbf{n}=(n_{1},\ldots,n_{m})$ samples at $(s_{1},\ldots,s_{m})$ with its $n-1$ coalescent transitions, so that $$\label{prior_rts}
P(g \mid {\textbf{t}}{},{\textbf{s}}{},{\textbf{n}})=\prod_{i=2}^{n} P\Big[({\textbf{a}}{(t_i)},b(t_i))\,\Big|\, ({\textbf{a}}{(t_i-)},b(t_i-))\Big].$$ Equation  gives the prior probability of the tree topology under the Tajima heterochronous $n$-coalescent. Putting together and , we obtain a prior $\pi({\textbf{g}}{}\mid {\textbf{s}}{},{\textbf{n}}{}, (N_e(t))_{t\geq 0})$ $$\label{prior}
\pi({\textbf{g}}{}\mid {\textbf{s}}{},{\textbf{n}}{}, (N_e(t))_{t\geq 0})=P(g \mid {\textbf{t}}{},{\textbf{s}}{},{\textbf{n}}) \prod_{k=3}^{n+1}p(t_{k-1}\mid t_{k},\mathbf{s},\mathbf{n},(N_{e}(t))_{t\geq 0}),$$ which can be used in .

Data and Likelihood {#lik}
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Infinite Sites Model and the Perfect Phylogeny {#data}
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We assume that the observed data [**Y**]{} consists of $n$ sequences 