Phylogeographic methods can help reveal the movement of genes between populations of organisms. This has been widely done to quantify pathogen movement between different host populations, the migration history of humans, and the geographic spread of languages or gene flow between species using the location or state of samples alongside sequence data. Phylogenies therefore offer insights into migration processes not available from classic epidemiological or occurrence data alone. Phylogeographic methods have however several known shortcomings. In particular, one of the most widely used methods treats migration the same as mutation, and therefore does not incorporate information about population demography. This may lead to severe biases in estimated migration rates for data sets where sampling is biased across populations. The structured coalescent on the other hand allows us to coherently model the migration and coalescent process, but current implementations struggle with complex data sets due to the need to infer ancestral migration histories. Thus, approximations to the structured coalescent, which integrate over all ancestral migration histories, have been developed. However, the validity and robustness of these approximations remain unclear. We present an exact numerical solution to the structured coalescent that does not require the inference of migration histories. Although this solution is computationally unfeasible for large data sets, it clarifies the assumptions of previously developed approximate methods and allows us to provide an improved approximation to the structured coalescent. We have implemented these methods in BEAST2, and we show how these methods compare under different scenarios.
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