14:00 – 15:00 – Federico Camerlenghi (University of Milano-Bicocca)
Title: Normalized random measures in Bayesian nonparametrics
Abstract: The seminal work of Ferguson (1973), who introduced the Dirichlet process, has spurred the definition and investigation of more general classes of Bayesian nonparametric priors, with the aim at increasing flexibility while maintaining analytical tractability. Among the numerous generalizations, a very large class of random probability measures have been introduced by Regazzini et al. (2003): this is the class of normalized random measures with independent increments (NRMIs). NRMIs are random probability measures with almost surely discrete realizations, defined through the specifications of two ingredients: i) a sequence of unnormalized weights, which are the jumps of a Levy process on the positive real line; ii) a sequence of i.i.d. random atoms from a common base measure. The proposed construction is appealing from a mathematical stand- point, because analytical tractability is preserved, however NRMIs do not allow interaction among atoms, which are supposed to be independent and identically distributed. In some applied framework, the i.i.d. assumption could be too restrictive, for instance, in model-based clustering, when they are used as mixing measures in mixture models. To overcome this limitation, we propose a new class of normalized random measures with atoms’ interaction. In our construction the atoms come from a finite point process, which is marked with i.i.d. positive weights. Thus, a new class of random probability measures is obtained by normalization. The desired interaction among atoms is then induced by a suitable choice of the law of the point process, which can create a repulsive or attractive behaviour. By means of Palm calculus, we are able to characterize marginal, predictive and posterior distributions for the proposed model. We specialize all our results for several choices of the finite point process, i.e., in the Poisson, Determinantal, Gibbs and Shot-Noise Cox case. This talk is based on a joint work with Raffaele Argiento, Mario Beraha and Alessandra Guglielmi.
Refreshments available between 15:00 – 15:30, Huxley Common Room (HXLY 549)