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Bayesian Multi-Model Frameworks - Properly Addressing Conceptual Uncertainty in Applied Modelling

Höge, Marvin

We use models to understand or predict a system. Often, there are multiple plausible but competing model concepts. Hence, modelling is associated with conceptual uncertainty, i.e., the question about proper handling of such model alternatives. For mathematical models, it is possible to quantify their plausibility based on data and rate them accordingly. Bayesian probability calculus offers several formal multi-model frameworks to rate models in a finite set and to quantify their conceptual uncertainty as model weights. These frameworks are Bayesian model selection and averaging (BMS/BMA), Pseudo-BMS/BMA and Bayesian Stacking. The goal of this dissertation is to facilitate proper utilization of these Bayesian multi-model frameworks. They follow different principles in model rating, which is why derived model weights have to be interpreted differently, too. These principles always concern the model setting, i.e., how the models in the set relate to one another and the true model of the system that generated observed data. This relation is formalized in model scores that are used for model weighting within each framework. The scores resemble framework-specific compromises between the ability of a model to fit the data and the therefore required model complexity. Hence, first, the scores are investigated systematically regarding their respective take on model complexity and are allocated in a developed classification scheme. This shows that BMS/BMA always pursues to identify the true model in the set, that Pseudo-BMS/BMA searches the model with largest predictive power despite none of the models being the true one, and that, on that condition, Bayesian Stacking seeks reliability in prediction by combining predictive distributions of multiple models. An application example with numerical models illustrates these behaviours and demonstrates which misinterpretations of model weights impend, if a certain framework is applied despite being unsuitable for the underlying model setting. Regarding applied modelling, first, a new setting is proposed that allows to identify a ``quasi-true'' model in a set. Second, Bayesian Bootstrapping is employed to take into account that rating of predictive capability is based on only limited data. To ensure that the Bayesian multi-model frameworks are employed properly and goal-oriented, a guideline is set up. With respect to a clearly defined modelling goal and the allocation of available models to the respective setting, it leads to the suitable multi-model framework. Aside of the three investigated frameworks, this guideline further contains an additional one that allows to identify a (quasi-)true model if it is composed of a linear combination of the model alternatives in the set. The gained insights enable a broad range of users in science practice to properly employ Bayesian multi-model frameworks in order to quantify and handle conceptual uncertainty. Thus, maximum reliability in system understanding and prediction with multiple models can be achieved. Further, the insights pave the way for systematic model development and improvement.