In statistics, the Bhattacharyya distance measures the similarity of two probability distributions. It is closely related to the Bhattacharyya coefficient which is a measure of the amount of overlap between two statistical samples or populations. It is not a metric, despite named a "distance", since it does not obey the triangle inequality. Definition For probability distributions and on the same domain , the Bhattacharyya distance is defined as where is the Bhattacharyya coefficient for discrete probability distributions. For continuous probability distributions, with and where and are the probability density functions, the Bhattacharyya coefficient is defined as . More generally, given two probability measures on a measurable space , let be a (sigma finite) measure such that and are absolutely continuous with respect to i.e. such that , and for probability density functions with respect to defined -almost everywhere. Such a measure, even such a probability measure, always exists, e.g. . Then define the Bhattacharrya measure on by It does not depend on the measure , for if we choose a measure such that and an other measure choice are absolutely continuous i.e. and , then , and similarly for . We then have . We finally define the Bhattacharyya coefficient . By the above, the quantity does not depend on , and by the Cauchy inequality . In particular if is absolutely continuous wrt to with Radon Nikodym derivative , then Properties and . does not obey the triangle inequality, though the Hellinger distance does. Let , , where is the normal distribution with mean and variance , then And in general, given two multivariate normal distributions , where Note that the first term is a squared Mahalanobis distance. Applications The Bhattacharyya coefficient quantifies the "closeness" of two random statistical samples. Given two sequences from distributions , bin them into buckets, and let the frequency of samples from in bucket be , and similarly for , then the sample Bhattacharyya coefficient is which is an estimator of . The quality of estimation depends on the choice of buckets; too few buckets would overestimate , while too many would underestimate.. A common task in classification is estimating the separability of classes. Up to a multiplicative factor, the squared Mahalanobis distance is a special case of the Bhattacharyya distance when the two classes are normally distributed with the same variances. When two classes have similar means but significantly different variances, the Mahalanobis