North America (dataset described by [@fro17]).

Both Bayesian and frequentist methods rely on the marginal likelihood computed by integrating over the space $\mathcal{G}_n$ of tree topologies with $n$ leaves, and over the space of branch lengths. That is: $$\label{marg}
P(\textbf{Y}\mid (N_e(t))_{t\geq 0}, \mu)= \int_{\mathbf{g} \in \mathcal{G}_n\times \mathbb{R}_+^{n-1}} P(\textbf{Y}\mid \mathbf{g}, (N_e(t))_{t\geq 0}, \mu) \,{{\rm d}}\pi (\mathbf{g}\mid (N_e(t))_{t\geq 0}),$$ where the $n-1$ random variables with values in $\mathbb{R}_+$ are the times between consecutive coalescence events in the genealogy, and $\pi(\cdot \mid(N_e(t))_{t\geq 0})$ denotes the probability distribution on $\mathcal{G}_n\times \mathbb{R}_+^{n-1}$ implied by the ancestral process as a function of the past population size trajectory $(N_e(t))_{t\geq 0}$. The genealogy [**g**]{} is an auxiliary variable introduced to compute $P(\textbf{Y}\mid \mathbf{g}, N_e(t), \mu)$ because direct calculation of the marginal $P(\textbf{Y}| N_e(t), \mu)$ is intractable. The prevailing consensus in the literature is to compute with $\pi$ defined as the Kingman-coalescent prior law on leaf-labeled genealogies (formally introduced in [@kingn82; @king82]). However, the cardinality of $\mathcal{G}_n$ grows superexponentially with $n$ ($|\mathcal{G}_n|=n!(n-1)!/2^{n-1}$), creating a computational bottleneck in the calculation of the integral .

![image](genealogy_intro-eps-converted-to){width=".5\textwidth"}

An alternative to the Kingman $n$-coalescent is to use a lower resolution coalescent process, known as the Tajima $n$-coalescent [@taj83; @sai15; @pal19]. While the state 