Dynamic density estimation with diffusive Dirichlet mixtures

Abstract

We introduce a new class of nonparametric prior distributions on the space of continuously varying densities, induced by Dirichlet process mixtures which diffuse in time. These select time-indexed random functions without jumps, whose sections are continuous or discrete distributions depending on the choice of kernel. The construction exploits the widely used stick-breaking representation of the Dirichlet process and induces the time dependence by replacing the stick-breaking components with one-dimensional Wright–Fisher diffusions. These features combine appealing properties of the model, inherited from the Wright–Fisher diffusions and the Dirichlet mixture structure, with great flexibility and tractability for posterior computation. The construction can be easily extended to multi-parameter GEM marginal states, which include, for example, the Pitman–Yor process. A full inferential strategy is detailed and illustrated on simulated and real data.