The inference of demographic history from genome data is hindered by a lack of efficient computational approaches. In particular, it has proven difficult to exploit the information contained in the distribution of genealogies across the genome. We have previously shown that the generating function (GF) of genealogies can be used to analytically compute likelihoods of demographic models from configurations of mutations in short sequence blocks (Lohse et al. 2011). Although the GF has a simple, recursive form, the size of such likelihood calculations explodes quickly with the number of individuals and applications of this framework have so far been mainly limited to small samples (pairs and triplets) for which the GF can be written down by hand. Here we investigate several strategies for exploiting the inherent symmetries of the coalescent. In particular, we show that the GF of genealogies can be decomposed into a set of equivalence classes which allows likelihood calculations from non-trivial samples. Using this strategy, we automated blockwise likelihood calculations for a general set of demographic scenarios in Mathematica. These histories may involve population size changes, continuous migration, discrete divergence and admixture between multiple populations. To give a concrete example, we calculate the likelihood for a model of isolation with migration (IM), assuming two diploid samples without phase and outgroup information. We demonstrate the new inference scheme with an analysis of two individual butterfly genomes from the sister species Heliconius melpomene rosina and Heliconius cydno.
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