Publication: Flexible Bayesian Nonparametric Priors and Bayesian Computational Methods
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2016-02-29
Defense date
2015-11-13
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Abstract
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.
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Estadística bayesiana, Estadística no paramétrica