On the Laplace Approximation as Model Selection Criterion for Gaussian Processes

Model selection aims to find the best model in terms of accuracy, interpretability or simplicity, preferably all at once.
In this work, we focus on evaluating model performance of Gaussian process models, i.e. finding a metric that provides the best trade-off between all those criteria.
While previous work considers metrics like the likelihood, AIC or dynamic nested sampling, they either lack performance or have significant runtime issues, which severely limits applicability.
We address these challenges by introducing multiple metrics based on the Laplace approximation, where we overcome a severe inconsistency occuring during naive application of the Laplace approximation.
Experiments show that our metrics are comparable in quality to the gold standard dynamic nested sampling without compromising for computational speed.
Our model selection criteria allow significantly faster and high quality model selection of Gaussian process models.