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 t
erms 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.
