Importance re-sampling {MCMC} for cross-validation in inverse problems

Abstract

This paper presents a methodology for 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 to build a model with which to make statements about climate, given sediment. One natural aspect of this is to examine model fit via 'inverse' cross-validation. We discuss the advantages of inverse cross-validation in Bayesian model assessment. In high-dimensional MCMC studies the inverse cross-validation exercise can be computationally burdensome. We propose a fast method involving very many low-dimensional MCMC runs, using Importance Re-sampling to reduce the dimensionality. We demonstrate that, in addition, the method is particularly suitable for exploring multimodal distributions. We illustrate our proposed methodology with simulation studies and the complex, high-dimensional, motivating palaeoclimate problem.