The Dirichlet serves as the conjugate prior for the Multinomial, meaning that the posterior θ 1 ...θ k |X 1 ...X n is again distributed as a Dirichlet (with updated parameters).
it is instructive to reparameterize the Dirichlet with k + 1 parameters: the parameters α ′ i give the mean of the distribution, and α 0 gives the precision of the distribution.