Warped density ratio estimation.

Estimating the ratio p/q between two probability density functions is a fundamental problem in machine learning. By training classifiers to distinguish between samples from p and q, the class-posterior probabilities can be used to estimate density ratios. For two distributions with a large discrepancy, this method often fails because classifiers are trained on trivial tasks. To address this, a common strategy is to construct a series of intermediate distributions between p and q, and estimate ratios between successive pairs. However, it involves the construction and estimation of multiple ratios, and lacks clear guidelines for selecting suitable schemes to improve accuracy. In this work, we propose Warped Density Ratio Estimation (WDRE), a novel method based on normalizing flows. By pushing both p and q toward analytically tractable targets, WDRE reformulates the original ratio into a product of two new ratios with reduced discrepancy. This approach avoids the need for a series of intermediate distributions, and can be implemented in both the data and latent spaces due to the invertibility of flows. Extensive experiments on synthetic and real-world tasks demonstrate the effectiveness of our proposed method.