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

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

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

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

is the Bhattacharya coefficient.^[1]

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

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

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