Statistics in Practice is an important international series of texts, which provide detailed coverage of statistical concepts, methods and worked case studies in specific fields of investigation and study.With sound motivation and many worked practical examples, the books show in down-to-earth terms how to select and use an appropriate range of statistical techniques in a particular practical field within each title's special topic area.The books provide statistical support for professionals and research workers across a range of employment fields and research environments. Subject areas covered include medicine and pharmaceutics; industry, finance and commerce; public services; the earth and environmental sciences, and so on.The books also provide support to students studying statistical courses applied to the above areas. The demand for graduates to be equipped for the work environment has led to such courses becoming increasingly prevalent at universities and colleges.It is our aim to present judiciously chosen and well-written workbooks to meet everyday practical needs. The feedback of views from readers will be most valuable to monitor the success of this aim.A complete list of titles in this series appears at the end of the volume.ii

Science-based simulation models are widely used to predict the behavior of complex physical systems. It is also common to use observations of the physical system to solve the inverse problem, that is, to learn about the values of parameters within the model, a process which is often called calibration. The main goal of calibration is usually to improve the predictive performance of the simulator but the values of the parameters in the model may also be of intrinsic scientific interest in their own right. In order to make appropriate use of observations of the physical system it is important to recognize model discrepancy, the difference between reality and the simulator output. We illustrate through a simple example that an analysis that does not account for model discrepancy may lead to biased and over-confident parameter estimates and predictions. The challenge with incorporating model discrepancy in statistical inverse problems is being confounded with calibration parameters, which will only be resolved with meaningful priors. For our simple example, we model the model-discrepancy via a Gaussian process and demonstrate that through accounting for model discrepancy our prediction within the range of data is correct. However, only with realistic priors on the model discrepancy do we uncover the true parameter values. Through theoretical arguments we show that these findings are typical of the general problem of learning about physical parameters and the underlying physical system using science-based mechanistic models.

SUMMARY Bayesian comparison of models is achieved simply by calculation of posterior probabilities of the models themselves. However, there are difficulties with this approach when prior information about the parameters of the various models is weak. Partial Bayes factors offer a resolution of the problem by setting aside part of the data as a training sample. The training sample is used to obtain an initial informative posterior distribution of the parameters in each model. Model comparison is then based on a Bayes factor calculated from the remaining data. Properties of partial Bayes factors are discussed, particularly in the context of weak prior information, and they are found to have advantages over other proposed methods of model comparison. A new variant of the partial Bayes factor, the fractional Bayes factor, is advocated on grounds of consistency, simplicity, robustness and coherence.

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