On the accuracy of posterior recovery with neural network emulators

Neural network emulators are widely used in astrophysics and cosmology to approximate complex simulations inside Bayesian inference loops. Ad hoc rules of thumb are often used to justify the emulator accuracy required for reliable posterior recovery. We provide a theoretically motivated limit on the maximum amount of incorrect information inferred by using an emulator with a given accuracy. Under assumptions of linearity in the model, uncorrelated noise in the data and a Gaussian likelihood function, we demonstrate that the difference between the true underlying posterior and the recovered posterior can be quantified via a Kullback-Leibler divergence. We demonstrate how this limit can be used in the field of 21-cm cosmology by comparing the posteriors recovered when fitting mock data sets generated with the 1D radiative transfer code ARES directly with the simulation code and separately with an emulator. This paper is partly in response to and builds upon recent discussions in the literature which call into question the use of emulators in Bayesian inference pipelines. Upon repeating some aspects of these analyses, we find these concerns quantitatively unjustified, with accurate posterior recovery possible even when the mean RMSE error for the emulator is approximately 20% of the magnitude of the noise in the data. For the purposes of community reproducibility, we make our analysis code public at this link https://github.com/htjb/validating_posteriors.