of bison in North America, and to reconstruct population size trajectory of SARS-CoV-2 from viral sequences collected in France and Germany.'
author:
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    Lorenzo Cappello$^1$[^1], Amandine Véber$^{2*}$, Julia A. Palacios$^{1,3*}$\
    $^1$Department of Statistics, Stanford University\
    $^2$CMAP, CNRS, École Polytechnique, I.P. Paris\
    $^3$Department of Biomedical Data Science, Stanford Medicine
title: '**The Tajima heterochronous $n$-coalescent: inference from heterochronously sampled molecular data**'
---

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[*Keywords:*]{} Bayesian nonparametric, Kingman $n$-coalescent, multi-resolution, ancient DNA, Gaussian process.

Introduction {#sec:intro}
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Statistical inference of evolutionary parameters from a sample of $n$ DNA sequences accounts for the dependence among samples and models observed variation through two stochastic processes: an ancestral process of the sample represented by a genealogy [**g**]{}, which depends on the *effective population size trajectory* $(N_e(t))_{t\geq 0}$, and a mutation process with a given set of parameters $\mu$ that, conditionally on [**g**]{}, models the phenomena that have given rise to the sequences. However, state-of-the-art methodologies are not scalable to the amount of data available because the latent space of genealogies lives in a high dimensional space. In this paper, we tackle the problem of scalability from a modeling perspective: we propose a new ancestral process for heterochronous data that dramatically reduces 