Estimation of the Branch Points of a Branching Diffusion Process

A solution to the problem of estimating the positions and times of the branch points of a Brownian-motion/Yule process, given the positions of all the particles at a particular time, is outlined. A likelihood approach is used, and it is shown that the solution involves maintaining a clear distinction between likelihood and conditional probability if difficulties over mathematical singularities are to be avoided. Several unsolved mathematical problems are encountered, and it is concluded that some simulation studies may be required for a complete solution. Questions of scientific inference which the problem raises are discussed briefly.