  We consider the problem of drawing samples from posterior distributions
formed under a Dirichlet prior and a truncated multinomial likelihood, by which
we mean a Multinomial likelihood function where we condition on one or more
counts being zero a priori. Sampling this posterior distribution is of interest
in inference algorithms for hierarchical Bayesian models based on the Dirichlet
distribution or the Dirichlet process, particularly Gibbs sampling algorithms
for the Hierarchical Dirichlet Process Hidden Semi-Markov Model. We provide a
data augmentation sampling algorithm that is easy to implement, fast both to
mix and to execute, and easily scalable to many dimensions. We demonstrate the
algorithm's advantages over a generic Metropolis-Hastings sampling algorithm in
several numerical experiments.
