Bhattacharyya angle
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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
- [math]\displaystyle{ \Delta(p,q) = \arccos \operatorname{BC}(p,q) }[/math]
where pi, qi are the probabilities assigned to the point i, for i = 1, ..., n, and
- [math]\displaystyle{ \operatorname{BC}(p,q) = \sum_{i=1}^n \sqrt{p_i q_i} }[/math]
is the Bhattacharya coefficient.[1]
The Bhattacharya distance is the geodesic distance in the orthant of the sphere [math]\displaystyle{ S^{n-1} }[/math] obtained by projecting the probability simplex on the sphere by the transformation [math]\displaystyle{ p_i \mapsto \sqrt{p_i},\ i=1,\ldots, n }[/math].
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
- [math]\displaystyle{ \Delta(\rho,\sigma) = \arccos \sqrt{F(\rho, \sigma)}. }[/math]
See also
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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