  Kernel density estimation (KDE) is integral to a range of generative and
discriminative tasks in machine learning. Drawing upon tools from the
multidimensional calculus of variations, we derive an optimal weight function
that reduces bias in standard kernel density estimates for density ratios,
leading to improved estimates of prediction posteriors and
information-theoretic measures. In the process, we shed light on some
fundamental aspects of density estimation, particularly from the perspective of
algorithms that employ KDEs as their main building blocks.
