Uncertainty Analysis for Random Computer Models

PROJECT SUMMARYAnalysis of mathematical models is one of the principle tools for studying complex systems. Uncertainty analysis for computer models of such systems is concerned with system calibration, i.e. learning about system inputs from system data, and system forecasting, i.e. learning about system outputs. Such analysis must address many sources of uncertainty, relating to imperfect knowledge of model input values, limited knowledge of the form of the model, discrepancies between the model and the system and errors in system data. These problems are particularly acute for high dimensional models which are expensive in computer time to evaluate. A general Bayesian approach has been developed which treats all of these uncertainties in a unified manner. As yet, the method is not well suited to analyse high dimensional random computer models, i.e. those which which, when evaluated repeatedly for the same input, will give different outputs. This project concerns methodology which will allow us to incorporate the treatment of such models within the standard Bayesian approach to computer modelling, while remaining tractable even for high dimensional input and output spaces. One of the key features of the general Bayesian approach, when we cannot fully evaluate the function of interest, is to express uncertainty about the function values using an emulator, which specifies such uncertainty by combining global regression modelling with residual Gaussian process forms. We shall develop emulators both for the full joint probability distributions produced by the computer model, and also for useful summary quantities from the distribution. In particular, for high dimensional problems, we will exploit the simplifications of the Bayes linear approach to retain tractability in our inferences. This approach will also exploit related developments concerning uncertainty analysis for multilevel models, as we can view sample size for given inputs as determining the accuracy level of the model. Given the emulator, we will develop methods for system calibration and forecasting, for a given collection of simulator evaluations and system data. We also address the design problem of choosing the values of the inputs, and the number of repetitions for each choice, at which we shall evaluate the function. Design will be batch sequential, moving the focus from designs for building the emulators to designs for testing the emulators to designs for solving the inferential problems for the system when the emulator is stable. The Bayes linear approach will be used to retain tractability for design calculations for high dimensions. The approach is general but we will focus on two substantial and important applications. The first arises in high energy physics where the aim is to improve understanding of the behaviour of subatomic particles based on models and data arising in high energy particle collisions. The second arises in systems biology where we are concerned with identification of large biochemical network models exploiting data based on gene expression levels.