Bayesian function learning using MCMC methods

The paper deals with the problem of reconstructing a continuous one-dimensional function from discrete noisy samples. The measurements may also be indirect in the sense that the samples may be the output of a linear operator applied to the function (linear inverse problem, deconvolution). In some cases, the linear operator could even contain unknown parameters that are estimated from a second experiment (joint identification-deconvolution problem). Bayesian estimation provides a unified treatment of this class of problems, but the practical calculation of posterior densities leads to analytically intractable integrals. In the paper it is shown that a rigourous Bayesian solution can be efficiently implemented by resorting to a MCMC (Markov chain Monte Carlo) simulation scheme. In particular, it is discussed how the structure of the problem can be exploited in order to improve computational and convergence performances. The effectiveness of the proposed scheme is demonstrated on two classical benchmark problems as well as on the analysis of IVGTT (IntraVenous Glucose Tolerance Test) data, a complex identification-deconvolution problem concerning the estimation of the insulin secretion rate following the administration of an intravenous glucose injection.