Bhattacharyya angle explained

In statistics, Bhattacharyya angle, also called statistical angle, is a measure of distance between two probability measures defined on a finite probability space. It is defined as

\Delta(p,q) = \arccos \operatorname{BC}(p,q)

where p_i, q_i are the probabilities assigned to the point i, for i = 1, ..., n, and

\operatorname{BC}(p,q) = \sum ^{n} _{i =1} \sqrt{p _{i} q _{i}}

is the Bhattacharya coefficient.^[1]

The Bhattacharya distance is the geodesic distance in the orthant of the sphere

S ^{n-1} obtained by projecting the probability simplex on the sphere by the transformation p _{i} \mapsto \sqrt{p _{i}, i=1,\ldots,} n .

This distance is compatible with Fisher metric. It is also related to Bures distance and fidelity between quantum states as for two diagonal states one has

\Delta(\rho,\sigma) = \arccos \sqrt{F(\rho, \sigma)}.

See also

• Bhattacharyya distance
• Hellinger distance

Notes and References

1. On a measure of divergence between two statistical populations defined by their probability distributions. Bhattacharya. Anil Kumar. 1943. Bulletin of the Calcutta Mathematical Society. 35. 99–109.