Hellinger distance

The Hellinger distance , also known as the Hellinger metric , is a metric for probability measures that are represented by probability densities . It is closely related to the total variation distance and allows, for example, to draw conclusions based on the distance between two probability measures as to whether these are singular to one another.

It was introduced in 1909 by Ernst Hellinger as part of functional analysis .

definition

Given are two probability measures and on the event space , both of which are absolutely continuous with regard to a σ-finite measure and thus have the density functions and with regard to the measure . The Hellinger distance is then defined as ${\ displaystyle P_ {1}}$ ${\ displaystyle P_ {2}}$ ${\ displaystyle (X, {\ mathcal {A}})}$ ${\ displaystyle \ mu}$ ${\ displaystyle f_ {1}}$ ${\ displaystyle f_ {2}}$ ${\ displaystyle \ mu}$

{\ displaystyle H (P_ {1}, P_ {2}) = \ left ({\ frac {1} {2}} \ int _ {X} | {\ sqrt {f_ {1}}} - {\ sqrt {f_ {2}}} | ^ {2} \ mathrm {d} \ mu \ right) ^ {\ tfrac {1} {2}}}

.

properties

It is always .

{\ displaystyle H (P_ {1}, P_ {2}) \ in [0,1]}

It is exactly if , that is, if the probability measures are singular to one another. ${\ displaystyle H (P_ {1}, P_ {2}) = 1}$ ${\ displaystyle P_ {1} \ perp P_ {2}}$

It is exactly when .

{\ displaystyle H (P_ {1}, P_ {2}) = 0}

{\ displaystyle P_ {1} = P_ {2}}

For products of probability measures applies

{\ displaystyle 1-H ^ {2} \ left (\ bigotimes _ {i = 1} ^ {n} P_ {i}, \ bigotimes _ {i = 1} ^ {n} Q_ {i} \ right) = \ prod _ {i = 1} ^ {n} (1-H ^ {2} (P_ {i}, Q_ {i}))}

.

This then follows for product dimensions

{\ displaystyle 1-H ^ {2} (P ^ {\ otimes n}, Q ^ {\ otimes n}) = (1-H ^ {2} (P, Q)) ^ {n}}

.

So product dimensions are asymptotically always singular or match.

If the total variation norm denotes, then applies ${\ displaystyle \ Vert \ cdot \ Vert _ {TV}}$

{\ displaystyle H ^ {2} (P_ {1}, P_ {2}) \ leq {\ tfrac {1} {2}} \ Vert P_ {1} -P_ {2} \ Vert _ {TV} \ leq {\ sqrt {2}} H (P_ {1}, P_ {2})}

.

Total variation norm and Hellinger distance are equivalent to each other, so they generate the same topology.

literature

Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .

Individual evidence

↑ Hellinger, New Justification of the Theory of Quadratic Forms of Infinitely Many Variables , Journal for Pure and Applied Mathematics, Volume 136, 1909, pp. 210-271

[1] Hellinger, New Justification of the Theory of Quadratic Forms of Infinitely Many Variables , Journal for Pure and Applied Mathematics, Volume 136, 1909, pp. 210-271