Bayesian3 Active Learning for the Gaussian Process Emulator Using Information Theory

Gaussian process emulators (GPE) are a machine learning approach that replicates computational demanding models using training runs of that model. Constructing such a surrogate is very challenging and, in the context of Bayesian inference, the training runs should be well invested. The current paper offers a fully Bayesian view on GPEs for Bayesian inference accompanied by Bayesian active learning (BAL). We introduce three BAL strategies that adaptively identify training sets for the GPE using information-theoretic arguments. The first strategy relies on Bayesian model evidence that indicates the GPE’s quality of matching the measurement data, the second strategy is based on relative entropy that indicates the relative information gain for the GPE, and the third is founded on information entropy that indicates the missing information in the GPE. We illustrate the performance of our three strategies using analytical- and carbon-dioxide benchmarks. The paper shows evidence of convergence against a reference solution and demonstrates quantification of post-calibration uncertainty by comparing the introduced three strategies. We conclude that Bayesian model evidence-based and relative entropy-based strategies outperform the entropy-based strategy because the latter can be misleading during the BAL. The relative entropy-based strategy demonstrates superior performance to the Bayesian model evidence-based strategy.

Bayesian model evidence as a practical alternative to deviance information criterion

While model evidence is considered by Bayesian statisticians as a gold standard for model selection (the ratio in model evidence between two models giving the Bayes factor), its calculation is often viewed as too computationally demanding for many applications. By contrast, the widely used deviance information criterion (DIC), a different measure that balances model accuracy against complexity, is commonly considered a much faster alternative. However, recent advances in computational tools for efficient multi-temperature Markov chain Monte Carlo algorithms, such as steppingstone sampling (SS) and thermodynamic integration schemes, enable efficient calculation of the Bayesian model evidence. This paper compares both the capability (i.e. ability to select the true model) and speed (i.e. CPU time to achieve a given accuracy) of DIC with model evidence calculated using SS. Three important model classes are considered: linear regression models, mixed models and compartmental models widely used in epidemiology. While DIC was found to correctly identify the true model when applied to linear regression models, it led to incorrect model choice in the other two cases. On the other hand, model evidence led to correct model choice in all cases considered. Importantly, and perhaps surprisingly, DIC and model evidence were found to run at similar computational speeds, a result reinforced by analytically derived expressions.

Model selection on solid ground: Rigorous comparison of nine ways to evaluate Bayesian model evidence

Bayesian model selection or averaging objectively ranks a number of plausible, competing conceptual models based on Bayes' theorem. It implicitly performs an optimal trade-off between performance in fitting available data and minimum model complexity. The procedure requires determining Bayesian model evidence (BME), which is the likelihood of the observed data integrated over each model's parameter space. The computation of this integral is highly challenging because it is as high-dimensional as the number of model parameters. Three classes of techniques to compute BME are available, each with its own challenges and limitations: (1) Exact and fast analytical solutions are limited by strong assumptions. (2) Numerical evaluation quickly becomes unfeasible for expensive models. (3) Approximations known as information criteria (ICs) such as the AIC, BIC, or KIC (Akaike, Bayesian, or Kashyap information criterion, respectively) yield contradicting results with regard to model ranking. Our study features a theory-based intercomparison of these techniques. We further assess their accuracy in a simplistic synthetic example where for some scenarios an exact analytical solution exists. In more challenging scenarios, we use a brute-force Monte Carlo integration method as reference. We continue this analysis with a real-world application of hydrological model selection. This is a first-time benchmarking of the various methods for BME evaluation against true solutions. Results show that BME values from ICs are often heavily biased and that the choice of approximation method substantially influences the accuracy of model ranking. For reliable model selection, bias-free numerical methods should be preferred over ICs whenever computationally feasible.