Joseph B. Nagel

Spectral likelihood expansions for Bayesian inference

A spectral approach to Bayesian inference is presented. It pursues the emulation of the posterior probability density. The starting point is a series expansion of the likelihood function in terms of orthogonal polynomials. From this spectral likelihood expansion all statistical quantities of interest can be calculated semi-analytically. The posterior is formally represented as the product of a reference density and a linear combination of polynomial basis functions. Both the model evidence and the posterior moments are related to the expansion coefficients. This formulation avoids Markov chain Monte Carlo simulation and allows one to make use of linear least squares instead. The pros and cons of spectral Bayesian inference are discussed and demonstrated on the basis of simple applications from classical statistics and inverse modeling.

Hamiltonian Monte Carlo and Borrowing Strength in Hierarchical Inverse Problems

AbstractBayesian approaches to uncertainty quantification and information acquisition in hierarchically defined inverse problems are presented. The techniques comprise simple updating, staged estimation, and multilevel model calibration. In particular, the estimation of material properties within an ensemble of identically manufactured structural elements is considered. It is shown how inferring the characteristics of an individual specimen can be accomplished by exhausting statistical strength from tests of other ensemble members. This is useful in experimental situations where evidence is scarce or unequally distributed. Hamiltonian Monte Carlo is proposed to cope with the numerical challenges of the devised approaches. The performance of the algorithm is studied and compared to classical Markov chain Monte Carlo sampling. It turns out that Bayesian posterior computations can be drastically accelerated.