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 p_i, q_i 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.^[1]

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 )}}.

References

↑ Bhattacharya, Anil Kumar (1943). "On a measure of divergence between two statistical populations defined by their probability distributions". Bulletin of the Calcutta Mathematical Society. 35: 99–109.

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Bhattacharyya angle

See also

References