  The use of hierarchical mixture priors with shared atoms has recently
flourished in the Bayesian literature for partially exchangeable data.
Leveraging on nested levels of mixtures, these models allow the estimation of a
two-layered data partition: across groups and across observations. This paper
discusses and compares the properties of such modeling strategies when the
mixing weights are assigned either a finite-dimensional Dirichlet distribution
or a Dirichlet process prior. Based on these considerations, we introduce a
novel hierarchical nonparametric prior based on a finite set of shared atoms, a
specification that enhances the flexibility of the induced random measures and
the availability of fast posterior inference. To support these findings, we
analytically derive the induced prior correlation structure and partially
exchangeable partition probability function. Additionally, we develop a novel
mean-field variational algorithm for posterior inference to boost the
applicability of our nested model to large multivariate data. We then assess
and compare the performance of the different shared-atom specifications via
simulation. We also show that our variational proposal is highly scalable and
that the accuracy of the posterior density estimate and the estimated partition
is comparable with state-of-the-art Gibbs sampler algorithms. Finally, we apply
our model to a real dataset of Spotify's song features, simultaneously
segmenting artists and songs with similar characteristics.
