Comparison of Numerical Approaches to Bayesian Updating

This paper investigates the Bayesian process of identifying unknown model parameters given prior information and a set of noisy measurement data. There are two approaches being adopted in this research: one that uses the classical formula for measures and probability densities and one that leaves the underlying measure unchanged and updates the relevant random variable. The former is numerically tackled by a Markov chain Monte Carlo procedure based on the Metropolis-Hastings algorithm, whereas the latter is implemented via the ensemble/square root ensemble Kalman filters, as well as the functional approximation approaches in the form of the polynomial chaos based linear Bayesian filter and its corresponding square root algorithm. The study attempts to show the principal differences between full and linear Bayesian updates when a direct or a transformed version of measurements are taken into consideration. In this regard the comparison of both strategies is provided on the example of a steady state diffusion equation with nonlinear and transformed linear measurement operators.