Bayesian noise model selection and system identification based on approximation of the evidence

The purpose of this work is to identify the parameters of a second order system from noisy data in a context where the difficulty is twofold. First, the model is strongly non linear and possibly non Gaussian. Second, the noise distribution is unknown. It is nevertheless assumed to belong to a finite set, thus, the identification issue is coupled with a model selection problem. In a Bayesian framework, both the selection and the estimation are optimally designed and finally based upon the posterior distribution for the model index and the parameters. Since the latter does not admit a closed-form expression, we resort to Markov Chain Monte Carlo techniques: one Gibbs sampler is run per noise model so as to approximate the evidence and then compute the posterior probability. The model is then selected and the parameters are estimated given the selected model. A first numerical assessment of the method is given based on simulated data.