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In data terms, we reduce the mutational configuration of a block k ¯ to k i ¯ = { k 1, k 2, …, k n − 1 } corresponding to counts of singletons, doubletons, etc. in each block.
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Lohse et al. (2011) showed how the generating function (GF) of genealogies can be used to derive the probability of mutational configurations for a large range of demographic models.
For each parameter combination, we simulated 1,000,000 loci to obtain the expected frequencies of mutational configurations for the bSFS scheme.
To get a sense of the extra information captured by the bSFS, we plotted the probability of mutational configurations (for n = 5 ).
Although our strategy of exploiting the symmetries of the coalescent by partitioning the GF of branch lengths into a sum over unlabeled tree shapes makes it possible to compute the probability of full mutational configurations for nontrivial sample sizes, in practice, the number of mutational configurations still explodes catastrophically for n > 5 (Lohse et al. 2015).
We compare the power of this new scheme to both the likelihood-based analysis for full mutational configurations (for small samples) and the genome-wide SFS and use simulations to investigate the sensitivity of the new method to intrablock recombination and population structure.
The probability of a mutational configuration is obtained by taking successive derivatives of the GF with respect to all relevant "dummy" variables, each corresponding to a different branch of the genealogy.
The general method for computing the probability of observing a particular mutational configuration, k ¯, which is defined as counts of mutations on t ¯ and can be interpreted as the likelihood, has been described in detail previously (Lohse et al. 2011, equation 1) and involves taking higher-order derivatives of the GF of genealogical branches.
Each possible combination of mutation counts is a unique mutational configuration.
Because we distinguish only n − 1 types of mutations, the number of mutational configurations goes down from (k m + 2 ) 2 (n − 1 ) to (k m + 2 ) (n − 1 ).
Here, we use the generating function of genealogies to derive the probability of mutational configurations in short sequence blocks under a simple bottleneck model.
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