Computational inference beyond Kingmans coalescent

Full likelihood inference under Kingmans coalescent is a computationally challenging problem to which importance sampling (IS) and the product of approximate conditionals (PAC) method have been applied successfully. Both methods can be expressed in terms of families of intractable conditional sampling distributions (CSDs), and rely on principled approximations for accurate inference. Recently, more general $Lambda$- and $Xi$-coalescents have been observed to provide better modelling fits to some genetic data sets. We derive families of approximate CSDs for finite sites $Lambda$- and $Xi$-coalescents, and use them to obtain approximately optimal IS and PAC algorithms for $Lambda$-coalescents, yielding substantial gains in efficiency over existing methods.