Importance sampling on coalescent histories. I

Stephens and Donnelly (2000) constructed an efficient sequential importance-sampling proposal distribution on coalescent histories of a sample of genes for computing the likelihood of a type configuration of genes in the sample. In the current paper a characterization of their importance-sampling proposal distribution is given in terms of the diffusion-process generator describing the distribution of the population gene frequencies. This characterization leads to a new technique for constructing importance-sampling algorithms in a much more general framework when the distribution of population gene frequencies follows a diffusion process, by approximating the generator of the process.