Making and evaluating predictions

Given a sequence of measurements, a statistical model is a proposed solution to the inverse problem. Statistical models can be used to produce probabilistic predictions of future measurements. This thesis provides an exposition of the process of making and evaluating probabilistic predictions, with an overview of statistical models. A review of statistical decision theory is given, and a statistical decision problem, arising in computational biology, is considered. The proposed solution to the decision problem is used to decide which genes, within a bacterial population, are labelled as core genes. The process of statistical modelling is understood as a decision problem, where the predictive density is the optimal action. The corresponding loss functions are known as proper scoring rules. Proper scoring rules can also be used for model parameter inference, and the large sample theory of the minimum score estimate is described. An application of proper scoring rules using the minimum score estimate is described and is shown to generalise classical model selection criteria. A further application analyses measurement simulated from a noisy dynamical system. Four different models are applied to measurements with measurement noise simulated in the perfect model scenario, and in the imperfect model scenario. A numerical approximation to the exact Bayesian model is shown to outperform the other models in the perfect model scenario, but performance in the imperfect model scenario is dismal. This suggests that complicated models may not perform well if the measurements are from a system that deviates too far from the model. It emerges that models should always include an evaluation based on predictive performance