## Filtering Poisson-Dirichlet diffusions

We tackle inference, in a hidden Markov model framework, for the trajectory of a one- or two-parameter Poisson-Dirichlet diffusion, which evolve in the infinite-dimensional ordered simplex and model the continuous random evolution of an infinite vector of ranked frequencies. The problem is particularly challenging in that the symmetry induced by ranking the frequencies makes the collected observations take the form of unlabeled partitions, where group types are not specified, and computing the likelihood requires integrating over all possible probabilistic generation of the observed partition given the underlying state of the random measure. Furthermore, the time-marginal model is not conjugate, as conditioning on further sets of data gives rise to mixtures of nonparametric laws whose cardinality grows exponentially, making the statistical problem virtually intractable. In this setting, we provide recursive formulae for the signal filtering distribution, i.e., the conditional law of the diffusion state given past and present observations, which are finite mixtures whose weights are given by the laws of certain coagulated partitions. We devise suitable approximation schemes that allow to efficiently implement the filter given partition-valued data, and similarly tackle the marginal smoothing problem, given also future data. Joint work with Marco Dalla Pria (University of Torino) and Dario Span\`o (University of Warwick).

Area: CS13 - Diffusion and coalescent processes in population genetics (Martina Favero)

Keywords: Hidden Markov model, population genetics, infinite-dimensional diffusion, ranked partitions

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