Dynamic density estimation with diffusive Dirichlet mixtures

We introduce a new class of nonparametric prior distributions on the space of continuously varying den- sities, 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 pro- cess 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 pos- terior 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 sim- ulated and real data.