Abstract
Asymptotic unbiasedness and L2-consistency are established, under mild conditions, for the estimates of the Kullback–Leibler divergence between two probability measures in Rd, absolutely continuous with respect to (w.r.t.) the Lebesgue measure. These estimates are based on certain k-nearest neighbor statistics for pair of independent identically distributed (i.i.d.) due vector samples. The novelty of results is also in treating mixture models. In particular, they cover mixtures of nondegenerate Gaussian measures. The mentioned asymptotic properties of related estimators for the Shannon entropy and cross-entropy are strengthened. Some applications are indicated.
Highlights
The Kullback–Leibler divergence introduced in [1] is used for quantification of similarity of two probability measures
The Kullback–Leibler divergence itself belongs to a class of f -divergence measures (with f (t) = log t)
Our goal is to provide wide conditions for the asymptotic unbiasedness and L2 consistency of the specified Kullback–Leibler divergence estimates without such smoothness and boundedness hypotheses
Summary
The Kullback–Leibler divergence introduced in [1] is used for quantification of similarity of two probability measures. It plays important role in various domains such as statistical inference (see, e.g., [2,3]), metric learning [4,5], machine learning [6,7], computer vision [8,9], network security [10], feature selection and classification [11,12,13], physics [14], biology [15], medicine [16,17], finance [18], among others.
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