Flexible Bayesian Nonparametric Priors and Bayesian Computational Methods

Resum

The definition of vectors of dependent random probability measures is a topic of interest in Bayesian nonparametrics. They represent dependent nonparametric prior distributions that are useful for modelling observables for which specific covariate values are known. Our first contribution is the introduction of novel multivariate vectors of two-parameter Poisson-Dirichlet process. The dependence is induced by applying a L´evy copula to the marginal L´evy intensities. Our attention particularly focuses on the derivation of the Laplace functional transform and the analytical expression of the Exchangeable Partition Probability function (EPPF). Their knowledge allows us to gain some insight on the dependence structure of the priors defined. The second part of the thesis deals with the definition of Bayesian nonparametric priors through the class of species sampling models. In particular, we focus on the novel Beta-GOS model introduced by Airoldi, Costa, et al. (2014). Our second contribution is the modification of the Beta-GOS model with the motivation to accommodate both temporal and spatial correlations that exist in many applications. We then apply the modified model to simulated fMRI data and display the results. Finally, we aim to give contribution to another popular area of nonparametric computational methods in Bayesian inference: Approximate Bayesian Computations (ABC), by providing a new sampler BCbl. It combines the idea of standard ABC and bootstrap likelihood and allows to avoid the choice of ABC parameters. Our work is actually inspired by a recent algorithm BCel proposed by Mengersen, Pudlo and Robert (2013) that uses the well-established empirical likelihood approximation. However, to ensure that the empirical likelihood converges to the true likelihood, it requires a very careful choice of the constraints. This choice is not clear in many cases. On the other hand, the bootstrap likelihood is an automatic procedure, with only a few trivial parameters to specify. The advantages of our algorithm BCbl are illustrated with several examples.