Summary:
In this diploma thesis we present the Bayesian Inference for Spatial Poisson
Hidden Markov Model in discrete time and finite state space, applying in
Spatial Statistics and Disease Mapping.
The first chapter is the introduction of Spatial Statistics. It is full of
definitions, concepts and references in order to understand the meaning on
Spatial Statistics and its branch.
Moreover, the second chapter has to do with the Bayesian Statistics. It
includes the main elements of Bayesian approach, applications in the mixing of
Poisson allocations, the technical data augmentation and the description of
algorithm Gibbs which is part of the MCMC algorithm family. Moreover the third
chapter describes HMMs (Hidden Markov Models) in constant and discrete event.
Also we try to describe and solve the three main problems which have to do with
the Statistical inference. The ‘Forward-Backward’ algorithm, the EM algorithm
and the MCMC algorithm are assessment methods which give us solutions for the
main inference problems for the Hidden Markov Models.
All these associated in chapter 4, in which we have a new concept, the
‘Disease Mapping’. This chapter has to do with Cartography science and Medical
science and how this applies and develops with the help of Bayesian statistics
in the branch of Mathematics. In the conclusion, the chapter 5 and 6 are the
main chapters of this thesis. In these chapters, we combine the theory of
Hidden Markov Models in the field of Disease Mapping, given a ‘Bayesian’
solution in the spread of disease, having results from the applications
occurred in France. It is an effort to predict the disease, and the
possibilities of spreading to neighboring areas.
Keywords:
Disease mapping, Finite mixture distributions, Hidden Markov Models, Poisson mixtures, Potts model
