In probability and statistics, the Hellinger distance (closely related to, although different from, the Bhattacharyya 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][2]

It is sometimes called the Jeffreys distance.^[3][4]

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 ${\displaystyle 1/2}$ 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 ${\displaystyle BC(P,Q)}$ as it can be defined as

${\displaystyle H(P,Q)={\sqrt {1-BC(P,Q)}}.}$

Hellinger distances are used in the theory of sequential and asymptotic statistics.^[5][6]

The squared Hellinger distance between two normal distributions ${\displaystyle P\sim {\mathcal {N}}(\mu _{1},\sigma _{1}^{2})}$ and ${\displaystyle Q\sim {\mathcal {N}}(\mu _{2},\sigma _{2}^{2})}$ is:

${\displaystyle H^{2}(P,Q)=1-{\sqrt {\frac {2\sigma _{1}\sigma _{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}}\,e^{-{\frac {1}{4}}{\frac {(\mu _{1}-\mu _{2})^{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}}.}$

The squared Hellinger distance between two multivariate normal distributions ${\displaystyle P\sim {\mathcal {N}}(\mu _{1},\Sigma _{1})}$ and ${\displaystyle Q\sim {\mathcal {N}}(\mu _{2},\Sigma _{2})}$ is ^[7]

${\displaystyle H^{2}(P,Q)=1-{\frac {\det(\Sigma _{1})^{1/4}\det(\Sigma _{2})^{1/4}}{\det \left({\frac {\Sigma _{1}+\Sigma _{2}}{2}}\right)^{1/2}}}\exp \left\{-{\frac {1}{8}}(\mu _{1}-\mu _{2})^{T}\left({\frac {\Sigma _{1}+\Sigma _{2}}{2}}\right)^{-1}(\mu _{1}-\mu _{2})\right\}}$

The squared Hellinger distance between two exponential distributions ${\displaystyle P\sim \mathrm {Exp} (\alpha )}$ and ${\displaystyle Q\sim \mathrm {Exp} (\beta )}$ is:

${\displaystyle H^{2}(P,Q)=1-{\frac {2{\sqrt {\alpha \beta }}}{\alpha +\beta }}.}$

The squared Hellinger distance between two Weibull distributions ${\displaystyle P\sim \mathrm {W} (k,\alpha )}$ and ${\displaystyle Q\sim \mathrm {W} (k,\beta )}$ (where ${\displaystyle k}$ is a common shape parameter and ${\displaystyle \alpha \,,\beta }$ are the scale parameters respectively):

${\displaystyle H^{2}(P,Q)=1-{\frac {2(\alpha \beta )^{k/2}}{\alpha ^{k}+\beta ^{k}}}.}$

The squared Hellinger distance between two Poisson distributions with rate parameters ${\displaystyle \alpha }$ and ${\displaystyle \beta }$, so that ${\displaystyle P\sim \mathrm {Poisson} (\alpha )}$ and ${\displaystyle Q\sim \mathrm {Poisson} (\beta )}$, is:

${\displaystyle H^{2}(P,Q)=1-e^{-{\frac {1}{2}}({\sqrt {\alpha }}-{\sqrt {\beta }})^{2}}.}$

The squared Hellinger distance between two beta distributions ${\displaystyle P\sim {\text{Beta}}(a_{1},b_{1})}$ and ${\displaystyle Q\sim {\text{Beta}}(a_{2},b_{2})}$ is:

${\displaystyle H^{2}(P,Q)=1-{\frac {B\left({\frac {a_{1}+a_{2}}{2}},{\frac {b_{1}+b_{2}}{2}}\right)}{\sqrt {B(a_{1},b_{1})B(a_{2},b_{2})}}}}$

where ${\displaystyle B}$ is the beta function.

The squared Hellinger distance between two gamma distributions ${\displaystyle P\sim {\text{Gamma}}(a_{1},b_{1})}$ and ${\displaystyle Q\sim {\text{Gamma}}(a_{2},b_{2})}$ is:

${\displaystyle H^{2}(P,Q)=1-\Gamma \left({\scriptstyle {\frac {a_{1}+a_{2}}{2}}}\right)\left({\frac {b_{1}+b_{2}}{2}}\right)^{-(a_{1}+a_{2})/2}{\sqrt {\frac {b_{1}^{a_{1}}b_{2}^{a_{2}}}{\Gamma (a_{1})\Gamma (a_{2})}}}}$

where ${\displaystyle \Gamma }$ is the gamma function.

The Hellinger distance ${\displaystyle H(P,Q)}$ and the total variation distance (or statistical distance) ${\displaystyle \delta (P,Q)}$ are related as follows:^[8]

${\displaystyle H^{2}(P,Q)\leq \delta (P,Q)\leq {\sqrt {2}}H(P,Q)\,.}$

The constants in this inequality may change depending on which renormalization you choose (${\displaystyle 1/2}$ or ${\displaystyle 1/{\sqrt {2}}}$).

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

• Statistical distance
• Kullback–Leibler divergence
• Bhattacharyya distance
• Total variation distance
• Fisher information metric