Inferência Bayesiana exata para processos de Cox level-set

Abstract

This thesis proposes a novel family of multidimensional Cox processes with piece-wise constant intensity function and an exact Bayesian approach to perform statistical inference in this family. This family is based on the Bayesian Level-set model proposed by Dunlop et al. [2016] and is motivated by the fact that such processes may be efficient to model a variety of point process phenomena. Furthermore, due to its simpler form when compared to continuously varying intensity functions, it is expected to provided more precise results. A level set function depends on a latent Gaussian process to flexibly determines the regions of the space with constant intensities. Despite the intractability of the likelihood function and infinite dimensionality of the parameter space, the proposed methodology does not resource to discrete approximations of the space (unlike competing methodologies in the literature) and Monte Carlo is the only source of inaccuracy. This arises from an MCMC algorithm that converges to the exact posterior distribution of all the unknown quantities in the model. The MCMC algorithm relies on recent stochastic simulation techniques, such as Pseudo-Marginal Metropolis and Poisson estimator. Finally simulated and real examples are presented to demonstrate the efficiency and applicability of the proposed methodology.