Advances in Markov chain Monte Carlo MCMC methods now make it
computationally feasible and relatively straightforward to apply the Dirichlet
process prior in a wide range of Bayesian nonparametric problems. The
feasibility of these methods rests heavily on the fact that the MCMC approach
avoids direct sampling of the Dirichlet process and is instead based on
sampling the finite-dimensional posterior which is obtained from marginalizing
out the process.
¶ In application, it is the integrated posterior that is used in the
Bayesian nonparametric inference, so one might wonder about its theoretical
properties. This paper presents some results in this direction. In particular,
we will focus on a study of the posterior’s asymptotic behavior,
specifically for the problem when the data is obtained from a finite
semiparametric mixture distribution. A complication in the analysis arises
because the dimension for the posterior, although finite, increases with the
sample size. The analysis will reveal general conditions that ensure
exponential posterior consistency for a finite dimensional parameter and which
can be slightly generalized to allow the unobserved nonparametric parameters to
be sampled from a generalized Pólya urn scheme. Several interesting
examples are considered.