The precision parameter plays an important role in the Dirichlet Process.
When assigning a Dirichlet Process prior to the set of probability measures
on Rk, k > 1, this can be restrictive in the sense that the variability is determined
by a single parameter. The aim of this paper is to construct an enrichment of
the Dirichlet Process that is more
exible with respect to the precision parameter
yet still conjugate, starting from the notion of enriched conjugate priors, which
have been proposed to address an analogous lack of
exibility of standard conjugate
priors in a parametric setting. The resulting enriched conjugate prior allows
more
exibility in modelling uncertainty on the marginal and conditionals. We
describe an enriched urn scheme which characterizes this process and show that it
can also be obtained from the stick-breaking representation of the marginal and
conditionals. For non atomic base measures, this allows global clustering of the
marginal variables and local clustering of the conditional variables. Finally, we
consider an application to mixture models that allows for uncertainty between
homoskedasticity and heteroskedasticity