Information‐theoretic scores for Bayesian model selection and similarity analysis : concept and application to a groundwater problem

Abstract

Bayesian model selection (BMS) and Bayesian model justifiability analysis (BMJ) provide a statistically rigorous framework for comparing competing models through the use of Bayesian model evidence (BME). However, a BME-based analysis has two main limitations: (a) it does not account for a model's posterior predictive performance after using the data for calibration and (b) it leads to biased results when comparing models that use different subsets of the observations for calibration. To address these limitations, we propose augmenting BMS and BMJ analyses with additional information-theoretic measures: expected log-predictive density (ELPD), relative entropy (RE) and information entropy (IE). Exploring the connection between Bayesian inference and information theory, we explicitly link BME and ELPD together with RE and IE to highlight the information flow in BMS and BMJ analyses. We show how to compute and interpret these scores alongside BME, and apply the framework to a controlled 2D groundwater setup featuring five models, one of which uses a subset of the data for calibration. Our results show how the information-theoretic scores complement BME by providing a more complete picture concerning the Bayesian updating process. Additionally, we demonstrate how both RE and IE can be used to objectively compare models that feature different data sets for calibration. Overall, the introduced Bayesian information-theoretic framework can lead to a better-informed decision by incorporating a model's post-calibration predictive performance, by allowing to work with different subsets of the data and by considering the usefulness of the data in the Bayesian updating process.