A fully Bayesian approach to assessment of model adequacy in inverse problems

We consider the problem of assessing goodness of fit of a single Bayesian model to the observed data in the inverse problem context. A novel procedure of goodness of fit test is proposed, based on construction of reference distributions using the 'inverse' part of the given model. This is motivated by an example from palaeoclimatology in which it is of interest to reconstruct past climates using information obtained from fossils deposited in lake sediment. Since climate influences species, the model is built in the forward sense, that is, fossils are assumed to depend upon climate. The model combines 'modern data' which consists of observed species composition and the corresponding observed climates with 'fossil data'; the latter data consisting of fossil species composition deposited in lake sediments for the past thousands of years, but the corresponding past climates are unknown. Interest focuses on prediction of unknown past climates, which is the inverse part of the model. Technically, given a model f (Y vertical bar X, theta), where Y is the observed data and X is a set of (non-random) covariates, we obtain reference distributions based on the posterior pi ((X) over tilde vertical bar Y), where (X) over tilde must be interpreted as the unobserved random vector corresponding to the observed covariates X. Put simply, if the posterior distribution pi ((X) over tilde vertical bar Y) gives high density to the observed covariates X, or equivalently, if the posterior distribution of T ((X) over tilde) gives high density to T (X), where T is any appropriate statistic, then we say that the model fits the data. Otherwise the model in question is not adequate. We provide decision-theoretic justification of our proposed approach and discuss other theoretical and computational advantages. We demonstrate our methodology with many simulated examples and three complex, high-dimensional, realistic palaeoclimate problems, including the motivating palaeoclimate problem. Although our proposal is ideally suited for checking model fit in inverse regression problems, we indicate that the proposal may be potentially extended for model checking in quite general Bayesian problems. However, we do not claim to have solved all issues involved; in fact, our aim in this paper is to discuss advantages of, and also to shed light on issues that could be potential future research topics. If nothing else, we hope to have been able to make a step forward in the right direction. (C) 2012 Elsevier B.V. All rights reserved.