• Source: 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




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 )}}.}





See also


Bhattacharyya distance
Hellinger distance




References

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