  In this paper we consider the problem of dynamic clustering, where cluster
memberships may change over time and clusters may split and merge over time,
thus creating new clusters and destroying existing ones. We propose a Bayesian
nonparametric approach to dynamic clustering via mixture modeling. Our approach
relies on a novel time-dependent nonparametric prior defined by combining: i) a
copula-based transformation of a Gaussian autoregressive process; ii) the
stick-breaking construction of the Dirichlet process. Posterior inference is
performed through a particle Markov chain Monte Carlo algorithm which is
simple, computationally efficient and scalable to massive datasets. Advantages
of the proposed approach include flexibility in applications, ease of
computations and interpretability. We present an application of our dynamic
Bayesian nonparametric mixture model to the study the temporal dynamics of
gender stereotypes in adjectives and occupations in the 20th and 21st centuries
in the United States. Moreover, to highlight the flexibility of our model we
present additional applications to time-dependent data with covariates and with
spatial structure.
