Statistical Analysis of Distance Estimators with Density Differences and Density Ratios

Estimating a discrepancy between two probability distributions from samples is an important task in statistics and machine learning. There are mainly two classes of discrepancy measures: distance measures based on the density difference, such as the Lp-distances, and divergence measures based on the density ratio, such as the φ-divergences. The intersection of these two classes is the L1-distance measure, and thus, it can be estimated either based on the density difference or the density ratio. In this paper, we first show that the Bregman scores, which are widely employed for the estimation of probability densities in statistical data analysis, allows us to estimate the density difference and the density ratio directly without separately estimating each probability distribution. We then theoretically elucidate the robustness of these estimators and present numerical experiments.