Savage-Dickey density ratio estimation with normalizing flows for Bayesian model comparison

A core motivation of science is to evaluate which scientific model best explains observed data. Bayesian model comparison provides a principled statistical approach to comparing scientific models and has found widespread application within cosmology and astrophysics. Calculating the Bayesian evidence is computationally challenging, especially as we continue to explore increasingly more complex models. The Savage-Dickey density ratio (SDDR) provides a method to calculate the Bayes factor (evidence ratio) between two nested models using only posterior samples from the super model. The SDDR requires the calculation of a normalised marginal distribution over the extra parameters of the super model, which has typically been performed using classical density estimators, such as histograms. Classical density estimators, however, can struggle to scale to high-dimensional settings. We introduce a neural SDDR approach using normalizing flows that can scale to settings where the super model contains a large number of extra parameters. We demonstrate the effectiveness of this neural SDDR methodology applied to both toy and realistic cosmological examples. For a field-level inference setting, we show that Bayes factors computed for a Bayesian hierarchical model (BHM) and simulation-based inference (SBI) approach are consistent, providing further validation that SBI extracts as much cosmological information from the field as the BHM approach. The SDDR estimator with normalizing flows is implemented in the open-source harmonic Python package.