Bayesian analysis of signals

Abstract

Extracting information from signals can be posed as an inverse problem. Since inverse problems are generally ill-posed, the problem is cast in a Bayesian setting. The objective is three-fold: First, estimation of the posterior distribution of model parameters from a given signal. Second, estimation of the posterior of the time shift between two signals. Third, estimation of the joint posterior distribution of a time shift and a model change given two different signals. The MAP solution can be obtained with approximate covariance around the MAP point by iterative search algorithms such as Gauss Newton (GN), and the full posterior solution can be obtained by stochastic processes such as Monte Carlo Markov Chain (MCMC) simulation. For small time- shifts, GN and MCMC yields similar results, making GN a favourable option in these cases as it is computationally much cheaper. Applying these methods to synthetic data using the convolutional model shows good correspondence between the inverted and true parameters for low to medium noise levels.