Likelihoods for a general class of ARGs under the SMC

Ancestral recombination graphs (ARGs) are the focus of much ongoing research interest. Recent progress in inference has made ARG-based approaches feasible across of range of applications, and many new methods using inferred ARGs as input have appeared. This progress on the long-standing problem of ARG inference has proceeded in two distinct directions. First, the Bayesian inference of ARGs under the Sequentially Markov Coalescent (SMC), is now practical for tens-to-hundreds of samples. Second, approximate models and heuristics can now scale to sample sizes two to three orders of magnitude larger. Although these heuristic methods are reasonably accurate under many metrics, one significant drawback is that the ARGs they estimate do not have the topological properties required to compute a likelihood under models such as the SMC under present-day formulations. In particular, heuristic inference methods typically do not estimate precise details about recombination events, which are currently required to compute a likelihood. In this paper we present a backwards-time formulation of the SMC and derive a straightforward definition of the likelihood of a general class of ARG under this model. We show that this formulation does not require precise details of recombination events to be estimated, and is robust to the presence of polytomies. We discuss the possibilities for inference that this opens.