Sequential piecewise PCE approximation of likelihood functions in Bayesian inference

Abstract

A method for sequentially constructing polynomial chaos expansions, with the aim of approximating likelihood functions that occur in Bayesian inference problems is presented. The proposed approach is called piecewise polynomial chaos expansion (P-PCE) and is based on sequentially constructing PCEs in refined domains on the residuals of previously constructed PCEs. The obtained local spectral representation allows the computation of posterior expectations by post-processing the PCE coefficients based on the recently developed concept of spectral likelihood expansion (SLE). This paper presents a summary of the proposed theory and showcases the solution of two Bayesian inference problems using the presented approach.