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Bhattacharyya Distance

For two discrete probability distributions P=(p_i) and Q=(q_i) on the same outcomes, the Bhattacharyya coefficient is

BC(P,Q)=sum_(i)sqrt(p_iq_i).
(1)

For probability density functions p(x) and q(x), it is

BC(P,Q)=intsqrt(p(x)q(x))dx.
(2)

The Bhattacharyya distance is then defined by

D_B(P,Q)=-lnBC(P,Q).
(3)

The coefficient lies between 0 and 1, attaining 1 for identical distributions. Consequently, the distance is nonnegative, may be infinite, and is zero for identical distributions. Although it is symmetric, it does not in general satisfy the triangle inequality and therefore is not a metric.

See also

Metric, Probability Density Function, Statistical Distribution, Triangle Inequality

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References

Bhattacharyya, A. "On a Measure of Divergence between Two Statistical Populations Defined by Their Probability Distributions." Bull. Calcutta Math. Soc. 35, 99-109, 1943.

Cite this as:

Weisstein, Eric W. "Bhattacharyya Distance." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BhattacharyyaDistance.html

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