  The two-parameter Poisson-Dirichlet diffusion is an infinite-dimensional
diffusion on the ordered simplex with a two-parameter Poisson-Dirichlet (alpha,
theta) stationary distribution. We derive a dual process representation for the
diffusion, suggested by Feng et al. (2011)'s spectral expansion of the
transition density, and its rearrangement by Zhou (2015). The dual process is
in terms of a line-of-descent process which tracks the evolution of non-mutant
frequencies from time zero. Remarkably the line-of-descent process does not
depend on alpha. Methods of proof use the sampling distribution of n points in
the two-parameter Poisson-Dirichlet diffusion. We connect the sampling
distribution with a generalized Blackwell and MacQueen Polya urn model.
