Inference in molecular population genetics

full likelihood-based inference for modern population genetics data present s methodological and computational challenges. The problem is of considerab le practical importance and has attracted recent attention, with the develo pment of algorithms based on importance sampling (IS) and Markov chain Mont e Carlo (MCMC) sampling. Here we introduce a new IS algorithm. The optimal proposal distribution for these problems can be characterized, and we explo it a detailed analysis of genealogical processes to develop a practicable a pproximation to it. We compare the new method with existing algorithms on a variety of genetic examples. Our approach substantially outperforms existi ng IS algorithms, with efficiency typically improved by several orders of m agnitude. The new method also compares favourably with existing MCMC method s in some problems, and less favourably in others, suggesting that both IS and MCMC methods have a continuing role to play in this area. We offer insi ghts into the relative advantages of each approach, and we discuss diagnost ics in the IS framework.