Английская Википедия:Bhattacharyya angle

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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

<math> \Delta(p,q) = \arccos \operatorname{BC}(p,q) </math>

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

<math> \operatorname{BC}(p,q) = \sum_{i=1}^n \sqrt{p_i q_i} </math>

is the Bhattacharya coefficient.

The Bhattacharya distance is the geodesic distance in the orthant of the sphere <math>S^{n-1}</math> obtained by projecting the probability simplex on the sphere by the transformation <math>p_i \mapsto \sqrt{p_i},\ i=1,\ldots, n</math>.

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

<math>\Delta(\rho,\sigma) = \arccos \sqrt{F(\rho, \sigma)}.</math>

See also

• Bhattacharyya distance
• Hellinger distance