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This paper provides a unified perspective for the Kullback-Leibler(KL)-divergence and the integral probability metrics (IPMs) from theperspective of maximum likelihood density-ratio estimation (DRE). Both theKL-divergence and the IPMs are widely used in various fields in applicationssuch as generative modeling. However, a unified understanding of these conceptshas still been unexplored. In this paper, we show that the KL-divergence andthe IPMs can be represented as maximal likelihoods differing only by samplingschemes, and use this result to derive a unified form of the IPMs and a relaxedestimation method. To develop the estimation problem, we construct anunconstrained maximum likelihood estimator to perform DRE with a stratifiedsampling scheme. We further propose a novel class of probability divergences,called the Density Ratio Metrics (DRMs), that interpolates the KL-divergenceand the IPMs. In addition to these findings, we also introduce someapplications of the DRMs, such as DRE and generative adversarial networks. Inexperiments, we validate the effectiveness of our proposed methods.

Unified Perspective on Probability Divergence via Maximum Likelihood  Density Ratio Estimation: Bridging KL-Divergence and Integral Probability  Metrics