  The Wright-Fisher model, originating in Wright (1931) is one of the canonical
probabilistic models used in mathematical population genetics to study how
genetic type frequencies evolve in time. In this paper we bound the rate of
convergence of the stationary distribution for a finite population
Wright-Fisher Markov chain with parent independent mutation to the Dirichlet
distribution. Our result improves the rate of convergence established in Gan et
al. (2017) from $O(1/\sqrt{N})$ to $O(1/N)$. The results are derived using
Stein's method, in particular, the prelimit generator comparison method.
