Bhattacharyya angle

Short description: Distance between two probability measures in statistics

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

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

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

$\mathrm{BC}(p,q)=\sum _{i=1}^n\sqrt{p_i{q}_i}$

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 $p_i\mapsto \sqrt{p_i},i=1,\dots ,n$.

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

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

See also

• Bhattacharyya distance
• Hellinger distance