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
Δ ( p , q ) = arccos BC ( p , q ) {\displaystyle \Delta (p,q)=\arccos \operatorname {BC} (p,q)}
where pi, qi are the probabilities assigned to the point i, for i=1,...,n, and
BC ( p , q ) = ∑ i = 1 n p i q i {\displaystyle \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 {\displaystyle S^{n-1}} obtained by projecting the probability simplex on the sphere by the transformation p i ↦ p i , i = 1 , … , n {\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
Δ ( ρ , σ ) = arccos F ( ρ , σ ) . {\displaystyle \Delta (\rho ,\sigma )=\arccos {\sqrt {F(\rho ,\sigma )}}.}