Human immunodeficiency virus (HIV) is a rapidly evolving pathogen that causes chronic infections, so genetic diversity within a single infection can be very high. High-throughput "deep" sequencing can now measure this diversity in unprecedented detail, particularly since it can be performed at different timepoints during an infection, and this offers a potentially powerful way to infer the evolutionary dynamics of the intra-host viral population. However, population genomic inference from HIV

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... quence data is challenging because of high rates of mutation and recombination, rapid demographic changes, and ongoing selective pressures. In this paper we develop a new method for inference using HIV deep sequencing data using an approach based on importance sampling of ancestral recombination graphs under a multi-locus coalescent model. The approach further extends recent progress in the approximation of so-called conditional sampling distributions, a quantity of key interest when approximating coalescent likelihoods. The chief novelties of our method are that it is able to infer rates of recombination and mutation, as well as the effective population size, while handling sampling over different timepoints and missing data without extra computational difficulty. We apply our method to a dataset of HIV-1, in which several hundred sequences were obtained from an infected individual at seven timepoints over two years. We find mutation rate and effective population size estimates to be comparable to those produced by the software BEAST. Additionally, our method is able to produce local recombination rate estimates. The software underlying our method, Coalescenator, is freely available.