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

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.Wir benutzen Modelle, um ein System zu verstehen oder vorherzusagen. Oft gibt es dabei mehrere plausible aber konkurrierende Modellkonzepte. Daher geht Modellierung einher mit konzeptioneller Unsicherheit, also der Frage nach dem angemessenen Umgang mit solchen Modellalternativen. Bei mathematischen Modellen ist es möglich, die Plausibilität jedes Modells anhand von Daten des Systems zu quantifizieren und Modelle entsprechend zu bewerten. Bayes'sche Wahrscheinlichkeitsrechnung bietet dazu verschiedene formale Multi-Modellrahmen, um Modellalternativen in einem endlichen Set zu bewerten und ihre konzeptionelle Unsicherheit als Modellgewichte zu beziffern. Diese Rahmen sind Bayes'sche Modellwahl und -mittelung (BMS/BMA), Pseudo-BMS/BMA und Bayes'sche Modellstapelung. Das Ziel dieser Dissertation ist es, den adäquaten Umgang mit diesen Bayes'schen Multi-Modellrahmen zu ermöglichen. Sie folgen unterschiedlichen Prinzipien in der Modellbewertung weshalb die abgeleiteten Modellgewichte auch unterschiedlich zu interpretieren sind. Diese Prinzipien beziehen sich immer auf das Modellsetting, also darauf, wie sich die Modelle im Set zueinander und auf das wahre Modell des Systems beziehen, welches bereits gemessene Daten erzeugt hat. Dieser Bezug ist in Kenngrößen formalisiert, die innerhalb jedes Rahmens der Modellgewichtung dienen. Die Kenngrößen stellen rahmenspezifische Kompromisse dar, zwischen der Fähigkeit eines Modells die Daten zu treffen und der dazu benötigten Modellkomplexität. Daher werden die Kenngrößen zunächst systematisch auf ihre jeweilige Bewertung von Modellkomplexität untersucht und in einem entsprechend entwickelten Klassifikationschema zugeordnet. Dabei zeigt sich, dass BMS/BMA stets verfolgt das wahre Modell im Set zu identifizieren, dass Pseudo-BMS/BMA das Modell mit der höchsten Vorsagekraft sucht, obwohl kein wahres Modell verfügbar ist, und dass Bayes'sche Modellstapelung unter dieser Bedingung Verlässlichkeit von Vorhersagen anstrebt, indem die Vorhersageverteilungen mehrerer Modelle kombiniert werden. Ein Anwendungsbeispiel mit numerischen Modellen verdeutlicht diese Verhaltenweisen und zeigt auf, welche Fehlinterpretationen der Modellgewichte drohen, wenn ein bestimmter Rahmen angewandt wird, obwohl er nicht zum zugrundeliegenden Modellsetting passt. Mit Bezug auf anwendungsorientierte Modellierung wird dabei erstens ein neues Setting vorgestellt, das es ermöglicht, ein ``quasi-wahres'' Modell in einem Set zu identifizieren. Zweitens wird Bayes'sches Bootstrapping eingesetzt um bei der Bewertung der Vorhersagegüte zu berücksichtigen, dass diese auf Basis weniger Daten erfolgt. Um zu gewährleisten, dass die Bayes'schen Multi-Modellrahmen angemessen und zielführend eingesetzt werden, wird schließlich ein Leitfaden erstellt. Anhand eines klar definierten Modellierungszieles und der Einordnung der gegebenen Modelle in das entspechende Setting leitet dieser zum geeigneten Multi-Modellrahmen. Neben den drei untersuchten Rahmen enthält dieser Leitfaden zudem einen weiteren, der es ermöglicht ein (quasi-)wahres Modell zu identifizieren, wenn dieses aus einer Linearkombination der Modellalternativen im Set besteht. Die gewonnenen Erkenntnisse ermöglichen es einer breiten Anwenderschaft in Wissenschaft und Praxis, Bayes'sche Multi-Modellrahmen zur Quantifizierung und Handhabung konzeptioneller Unsicherheit adäquat einzusetzen. Dadurch lässt sich maximale Verlässlichkeit in Systemverständis und -vorhersage durch mehrere Modelle erreichen. Die Erkenntnisse ebnen darüber hinaus den Weg für systematische Modellentwicklung und -verbesserung

Bayesian Model Weighting: The Many Faces of Model Averaging

Model averaging makes it possible to use multiple models for one modelling task, like predicting a certain quantity of interest. Several Bayesian approaches exist that all yield a weighted average of predictive distributions. However, often, they are not properly applied which can lead to false conclusions. In this study, we focus on Bayesian Model Selection (BMS) and Averaging (BMA), Pseudo-BMS/BMA and Bayesian Stacking. We want to foster their proper use by, first, clarifying their theoretical background and, second, contrasting their behaviours in an applied groundwater modelling task. We show that only Bayesian Stacking has the goal of model averaging for improved predictions by model combination. The other approaches pursue the quest of finding a single best model as the ultimate goal, and use model averaging only as a preliminary stage to prevent rash model choice. Improved predictions are thereby not guaranteed. In accordance with so-called M -settings that clarify the alleged relations between models and truth, we elicit which method is most promising

Strategies for Simplifying Reactive Transport Models: A Bayesian Model Comparison

For simulating reactive transport on aquifer scale, various modeling approaches have been proposed. They vary considerably in their computational demands and in the amount of data needed for their calibration. Typically, the more complex a model is, the more data are required to sufficiently constrain its parameters. In this study, we assess a set of five models that simulate aerobic respiration and denitrification in a heterogeneous aquifer at quasi steady state. In a probabilistic framework, we test whether simplified approaches can be used as alternatives to the most detailed model. The simplifications are achieved by neglecting processes such as dispersion or biomass dynamics, or by replacing spatial discretization with travel‐time‐based coordinates. We use the model justifiability analysis proposed by Schöniger, Illman, et al. (2015, https://doi.org/10.1016/j.jhydrol.2015.07.047) to determine how similar the simplified models are to the reference model. This analysis rests on the principles of Bayesian model selection and performs a tradeoff between goodness‐of‐fit to reference data and model complexity, which is important for the reliability of predictions. Results show that, in principle, the simplified models are able to reproduce the predictions of the reference model in the considered scenario. Yet, it became evident that it can be challenging to define appropriate ranges for effective parameters of simplified models. This issue can lead to overly wide predictive distributions, which counteract the apparent simplicity of the models. We found that performing the justifiability analysis on the case of model simplification is an objective and comprehensive approach to assess the suitability of candidate models with different levels of detail.Plain Language Summary: In groundwater, chemical substances like nitrate are transported and undergo chemical reactions. Understanding such reactive transport processes plays a key role in securing our water resources and drinking water. We use computer models for understanding such reactive transport processes and for simulating their future behavior. In such models, we make many scientific decisions on which processes should be included and in what degree of detail. Here, we face a trade‐off: Usually, a complex model with many mathematical terms resolves many details of the process. Yet, such complex models require lots of data for calibration and lots of time for the computer simulation. In contrast, a simple model with fewer details comes with less effort in both respects. However, it might neglect important parts of the process. For the example of nitrate decay, we use a probabilistic approach to find the best simplification for a comparatively detailed reference model. Our results show that, in certain cases, it is justified to employ a simpler model instead of a complex alternative without deteriorating modeling results. Alongside, we explain how difficult it can be to define realistic parameter ranges for simplified models.Key Points: We compare a set of four simplified models against a reference model for reactive transport at quasi steady state on aquifer scale. A Bayesian model justifiability analysis helps identifying the most suitable model simplification strategy. The proposed analysis reveals the difficulty of reasonably constraining parameter priors for simplified models.DFG http://dx.doi.org/10.13039/50110000165