A finite-infinite shared atoms nested model for the Bayesian analysis of large grouped data

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.