Importance resampling MCMC : a methodology for cross-validation in inverse problems and its applications in model assessment

Abstract

This thesis presents a methodology for implementing cross-validation in the context of Bayesian modelling of situations we loosely refer to as 'inverse problems'. It is motivated by an example from palaeoclimatology in which scientists reconstruct past climates from fossils in lake sediment. The inverse problem is then to build a model with which to make statem ents about climate, given sediment. One natural aspect of this is to examine model fit via cross-validation. In MCMC studies this can be computationally burdensome and our procedure has attractive properties in this respect. We demonstrate that, in addition, it is possible to take advantage of the flexibility inherent within the method to make it suitable for exploring multimodal distributions. We also propose to construct useful reference distributions using data obtained from cross-validation. Our proposals are illustrated using a simulated data set and several real data sets.