Bhattacharyya angle

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 pi, qi are the probabilities assigned to the point i, for i = 1, ..., n, and


 * $$ \operatorname{BC}(p,q) = \sum_{i=1}^n \sqrt{p_i q_i} $$

is the Bhattacharya coefficient.

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


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