# Direct Density-Derivative Estimation and Its Application in KL-Divergence Approximation

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Hiroaki Sasaki,
Yung-Kyun Noh,
Masashi Sugiyama
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Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, PMLR 38:809-818, 2015.

#### Abstract

Estimation of density derivatives is a versatile tool in statistical data analysis. A naive approach is to first estimate the density and then compute its derivative. However, such a two-step approach does not work well because a good density estimator does not necessarily mean a good density-derivative estimator. In this paper, we give a direct method to approximate the density derivative without estimating the density itself. Our proposed estimator allows analytic and computationally efficient approximation of multi-dimensional high-order density derivatives, with the ability that all hyper-parameters can be chosen objectively by cross-validation. We further show that the proposed density-derivative estimator is useful in improving the accuracy of non-parametric KL-divergence estimation via metric learning. The practical superiority of the proposed method is experimentally demonstrated in change detection and feature selection.

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