Bhattacharyya angle

{{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

: $\backslash Delta(p,q)\; =\; \backslash arccos\; \backslash operatorname\{BC\}(p,q)$

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

: $\backslash operatorname\{BC\}(p,q)\; =\; \backslash sum\_\{i=1\}^n\; \backslash sqrt\{p\_i\; q\_i\}$

is the Bhattacharya coefficient.{{Cite journal|title = On a measure of divergence between two statistical populations defined by their probability distributions|last = Bhattacharya|first = Anil Kumar|date = 1943|journal = Bulletin of the Calcutta Mathematical Society|volume = 35|pages = 99–109}}

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 $p\_i\; \backslash mapsto\; \backslash sqrt\{p\_i\},\backslash \; i=1,\backslash ldots,\; n$.

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

: $\backslash Delta(\backslash rho,\backslash sigma)\; =\; \backslash arccos\; \backslash sqrt\{F(\backslash rho,\; \backslash sigma)\}.$