Brownian models and coalescent structures

Brownian motions on coalescent structures have a biological relevance, either as an approximation of the stepwise mutation model for microsatellites, or as a model of spatial evolution considering the locations of individuals at successive generations. We discuss estimation procedures for the dispersal parameter of a Brownian motion defined on coalescent trees. First, we consider the mean square distance unbiased estimator and compute its variance. In a second approach, we introduce a phylogenetic estimator. Given the UPGMA topology, the likelihood of the parameter is computed thanks to a new dynamical programming method. By a proper correction, an unbiased estimator is derived from the pseudomaximum of the likelihood. The last approach consists of computing the likelihood by a Markov chain Monte Carlo sampling method. In the one-dimensional Brownian motion, this method seems less reliable than pseudomaximum-likelihood.