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ⓘ Hellinger distance. In probability and statistics, the Hellinger distance is used to quantify the similarity between two probability distributions. It is a type ..




                                     

ⓘ Hellinger distance

In probability and statistics, the Hellinger distance is used to quantify the similarity between two probability distributions. It is a type of f -divergence. The Hellinger distance is defined in terms of the Hellinger integral, which was introduced by Ernst Hellinger in 1909.

                                     

1.1. Definition Measure theory

To define the Hellinger distance in terms of measure theory, let P and Q denote two probability measures that are absolutely continuous with respect to a third probability measure λ. The square of the Hellinger distance between P and Q is defined as the quantity

H 2 P, Q = 1 2 ∫ d P d λ − d Q d λ 2 d λ. {\displaystyle H^{2}P,Q={\frac {1}{2}}\displaystyle \int \left{\sqrt {\frac {dP}{d\lambda }}}-{\sqrt {\frac {dQ}{d\lambda }}}\right^{2}d\lambda.}

Here, dP / dλ and dQ / d λ are the Radon–Nikodym derivatives of P and Q respectively. This definition does not depend on λ, so the Hellinger distance between P and Q does not change if λ is replaced with a different probability measure with respect to which both P and Q are absolutely continuous. For compactness, the above formula is often written as

H 2 P, Q = 1 2 ∫ d P − d Q 2. {\displaystyle H^{2}P,Q={\frac {1}{2}}\int \left{\sqrt {dP}}-{\sqrt {dQ}}\right^{2}.}
                                     

1.2. Definition Probability theory using Lebesgue measure

To define the Hellinger distance in terms of elementary probability theory, we take λ to be Lebesgue measure, so that dP / dλ and dQ / d λ are simply probability density functions. If we denote the densities as f and g, respectively, the squared Hellinger distance can be expressed as a standard calculus integral

H 2 f, g = 1 2 ∫ f x − g x) 2 d x = 1 − ∫ f x g x d x, {\displaystyle H^{2}f,g={\frac {1}{2}}\int \left{\sqrt {fx}}-{\sqrt {gx}}\right)^{2}\,dx=1-\int {\sqrt {fxgx}}\,dx,}

where the second form can be obtained by expanding the square and using the fact that the integral of a probability density over its domain equals 1.

The Hellinger distance H P, Q satisfies the property derivable from the Cauchy–Schwarz inequality

0 ≤ H P, Q ≤ 1. {\displaystyle 0\leq HP,Q\leq 1.}
                                     

1.3. Definition Discrete distributions

For two discrete probability distributions P = p 1, …, p k {\displaystyle P=p_{1},\ldots,p_{k}} and Q = q 1, …, q k {\displaystyle Q=q_{1},\ldots,q_{k}}, their Hellinger distance is defined as

H P, Q = 1 2 ∑ i = 1 k p i − q i 2, {\displaystyle HP,Q={\frac {1}{\sqrt {2}}}\;{\sqrt {\sum _{i=1}^{k}{\sqrt {p_{i}}}-{\sqrt {q_{i}}}^{2}}},}

which is directly related to the Euclidean norm of the difference of the square root vectors, i.e.

H P, Q = 1 2 ‖ P − Q ‖ 2. {\displaystyle HP,Q={\frac {1}{\sqrt {2}}}\;{\bigl \|}{\sqrt {P}}-{\sqrt {Q}}{\bigr \|}_{2}.}

Also, 1 − H 2 P, Q = ∑ i = 1 k p i q i. {\displaystyle 1-H^{2}P,Q=\sum _{i=1}^{k}{\sqrt {p_{i}q_{i}}}.}

                                     

2. Connection with the statistical distance

The Hellinger distance H P, Q {\displaystyle HP,Q} and the total variation distance or statistical distance δ P, Q {\displaystyle \delta P,Q} are related as follows:

H 2 P, Q ≤ δ P, Q ≤ 2 H P, Q. {\displaystyle H^{2}P,Q\leq \delta P,Q\leq {\sqrt {2}}HP,Q\.}

These inequalities follow immediately from the inequalities between the 1-norm and the 2-norm.

                                     

3. Properties

The Hellinger distance forms a bounded metric on the space of probability distributions over a given probability space.

The maximum distance 1 is achieved when P assigns probability zero to every set to which Q assigns a positive probability, and vice versa.

Sometimes the factor 1 / 2 {\displaystyle 1/2} in front of the integral is omitted, in which case the Hellinger distance ranges from zero to the square root of two.

The Hellinger distance is related to the Bhattacharyya coefficient B C P, Q {\displaystyle BCP,Q} as it can be defined as

H P, Q = 1 − B C P, Q. {\displaystyle HP,Q={\sqrt {1-BCP,Q}}.}

Hellinger distances are used in the theory of sequential and asymptotic statistics.

The squared Hellinger distance between two normal distributions P ∼ N μ 1, σ 1 2 {\displaystyle \scriptstyle P\,\sim \,{\mathcal {N}}\mu _{1},\sigma _{1}^{2}} and Q ∼ N μ 2, σ 2 {\displaystyle \scriptstyle Q\,\sim \,{\mathcal {N}}\mu _{2},\sigma _{2}^{2}} is:

H 2 P, Q = 1 − 2 σ 1 σ 2 σ 1 2 + σ 2 e − 1 4 μ 1 − μ 2 σ 1 2 + σ 2 2. {\displaystyle H^{2}P,Q=1-{\sqrt {\frac {2\sigma _{1}\sigma _{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}}\,e^{-{\frac {1}{4}}{\frac {\mu _{1}-\mu _{2}^{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}}.}

The squared Hellinger distance between two multivariate normal distributions P ∼ N μ 1, ∑ 1 {\displaystyle \scriptstyle P\,\sim \,{\mathcal {N}}\mu _{1},\sum _{1}} and Q ∼ N μ 2, ∑ 2 {\displaystyle \scriptstyle Q\,\sim \,{\mathcal {N}}\mu _{2},\sum _{2}} is:

H 2 P, Q = 1 − det ∑ 1 / 4 det ∑ 2 1 / 4 det ∑ 1 + ∑ 2 1 / 2 exp ⁡ { − 1 8 μ 1 − μ 2 T ∑ 1 + ∑ 2 − 1 μ 1 − μ 2 } {\displaystyle H^{2}P,Q=1-{\frac {\det\sum _{1}^{1/4}\det\sum _{2}^{1/4}}{\det \left{\frac {\sum _{1}+\sum _{2}}{2}}\right^{1/2}}}\exp \left\{-{\frac {1}{8}}\mu _{1}-\mu _{2}^{T}\left{\frac {\sum _{1}+\sum _{2}}{2}}\right^{-1}\mu _{1}-\mu _{2}\right\}}

The squared Hellinger distance between two exponential distributions P ∼ E x p α {\displaystyle \scriptstyle P\,\sim \,{\rm

where B {\displaystyle B} is the Beta function.



                                     
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