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.

History 
Both the Bhattacharyya distance and the Bhattacharyya coefficient are named after Anil Kumar Bhattacharyya, a statistician who worked in the 1930s at the Indian Statistical Institute. He developed the method to measure the distance between two non-normal distributions and illustrated this with the classical multinomial populations as well as probability distributions that are absolutely continuous with respect to the Lebesgue measure. The latter work appeared partly in 1943 in the Bulletin of the Calcutta Mathematical Society, while the former part, despite being submitted for publication in 1941, appeared almost five years later in Sankhya.

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 Bhattacharyya 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 distance would be close to zero, while the Bhattacharyya distance would not be.

The Bhattacharyya coefficient is used in the construction of polar codes.

The Bhattacharyya distance is used in feature extraction and selection, image processing, speaker recognition, and phone clustering.

See also 

 Bhattacharyya angle
 Kullback–Leibler divergence
 Hellinger distance
 Mahalanobis distance
 Chernoff bound
 Rényi entropy
 F-divergence
 Fidelity of quantum states

References 

 

 

 

 For a short list of properties, see: http://www.mtm.ufsc.br/~taneja/book/node20.html

External links
 
 Bhattacharyya's distance measure as a precursor of genetic distance measures, Journal of Biosciences, 2004
 Statistical Intuition of Bhattacharyya's distance

Statistical distance
Statistical deviation and dispersion
Anil Kumar Bhattacharya