Distance de Hellinger

In probability and statistics, the Hellinger distance (closely related to, although different from, the Bhattacharyya distance) is used to quantify the similarity between two probability distributions. It is a type of f-divergence. The Hellinger distance is defined in terms of the Hellinger integral, which was introduced by Ernst Hellinger in 1909. It is sometimes called the Jeffreys distance. To define the Hellinger distance in terms of measure theory, let and denote two probability measures on a measure space that are absolutely continuous with respect to an auxiliary measure . Such a measure always exists, e.g . The square of the Hellinger distance between and is defined as the quantity Here, and , i.e. and are the Radon–Nikodym derivatives of P and Q respectively with respect to . This definition does not depend on , i.e. the Hellinger distance between P and Q does not change if is replaced with a different probability measure with respect to which both P and Q are absolutely continuous. For compactness, the above formula is often written as To define the Hellinger distance in terms of elementary probability theory, we take λ to be the Lebesgue measure, so that dP / dλ and dQ / dλ are simply probability density functions. If we denote the densities as f and g, respectively, the squared Hellinger distance can be expressed as a standard calculus integral where the second form can be obtained by expanding the square and using the fact that the integral of a probability density over its domain equals 1. The Hellinger distance H(P, Q) satisfies the property (derivable from the Cauchy–Schwarz inequality) For two discrete probability distributions and , their Hellinger distance is defined as which is directly related to the Euclidean norm of the difference of the square root vectors, i.e. Also, The Hellinger distance forms a bounded metric on the space of probability distributions over a given probability space. The maximum distance 1 is achieved when P assigns probability zero to every set to which Q assigns a positive probability, and vice versa.