MCCM methods for Bayesian analysis of nonlinear parametric regression models. Application to line analysis and impulse deconvolution.

Summary In this thesis, the general linear regression problem, which involves non-linear parameters, is addressed in a Bayesian framework. This approach allows theorizing the difficult problems of nonlinear parameter estimation as well as the choice of the model allowing a parsimonious representation of the observed signal. The effective implementation of Bayesian statistics requires the use of numerical procedures. The procedures used in this work are monte carlo methods by markov chains (mcmc) which allow to efficiently carry out integration and optimization on a union of spaces of different dimensions. The procedures developed are applied to the problem of spectral analysis of sinusoids embedded in white Gaussian noise. A monte carlo study of the performances of different model selection procedures, derived from the developed algorithms and from classical criteria, is presented. We then show how it is possible to extend the previously proposed procedures to cases where the observation noise can be non-Gaussian or colored. We also show how these algorithms can be applied when the data undergo a thresholding phenomenon, preventing the observation of the noisy process beyond certain thresholds. The problem of deconvolution of continuous-time filtered and noisy point processes is also addressed in a Bayesian framework and solved by means of mcmc methods. The statistical model and the associated algorithms are applied to real spectrometry data.