  Exchangeable random partition processes are the basis for Bayesian approaches
to statistical inference in large alphabet settings. On the other hand, the
notion of the pattern of a sequence provides an information-theoretic framework
for data compression in large alphabet scenarios. Because data compression and
parameter estimation are intimately related, we study the redundancy of Bayes
estimators coming from Poisson-Dirichlet priors (or "Chinese restaurant
processes") and the Pitman-Yor prior. This provides an understanding of these
estimators in the setting of unknown discrete alphabets from the perspective of
universal compression. In particular, we identify relations between alphabet
sizes and sample sizes where the redundancy is small, thereby characterizing
useful regimes for these estimators.
