Learning Gaussian Conditional Distributions using Neural Ratio Estimation is Hard

Neural Ratio Estimation (NRE) is a popular method for performing simulation-based posterior estimation given a known prior and samples from a joint distribution. For such a task, it can be crucial for the posterior estimation task to be \emph{statistically efficient}, i.e. to require as few simulations as possible to obtain an accurate posterior estimate. However, currently, little is known about its statistical efficiency. In this work, we bridge this gap by performing an asymptotic statistical analysis of NRE. We show that even when the target distribution is a simple product of Gaussians, the accuracy of NRE can degrade exponentially badly with the dimension of the data distribution. Our results show that NRE can be significantly less efficient than other conditional density methods like Maximum Likelihood Estimation, which are known to enjoy favorable statistical properties.