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ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
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ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
This module contains classes and procedures for computing the Hellinger statistical distance between two probability distributions. More...
Data Types | |
| interface | getDisHellSq |
| Generate and return the square of the Hellinger distance of two univariate (discrete or continuous) distributions. More... | |
Variables | |
| character(*, SK), parameter | MODULE_NAME = "@pm_distanceHellinger" |
This module contains classes and procedures for computing the Hellinger statistical distance between two probability distributions.
The Hellinger distance (which is also closely related to the Bhattacharyya distance) is used to quantify the similarity between two probability distributions.
The Hellinger distance is defined in terms of the Hellinger integral, which was introduced by Ernst Hellinger in 1909.
It is also sometimes called the Jeffreys distance.
Let \(P\) and \(Q\) denote two probability measures on a measure space \(\mathcal{X}\) that are absolutely continuous with respect to an auxiliary measure \(\lambda\).
The square of the Hellinger distance between \(P\) and \(Q\) is defined as the quantity,
\begin{equation} H^{2}(P, Q) = {\frac{1}{2}} \int_{\mathcal{X}} \left( {\sqrt{p(x)}} - {\sqrt {q(x)}} \right)^{2} \lambda(dx) ~. \end{equation}
where, \(P(dx) = p(x)\lambda(dx)\) and \(Q(dx) = q(x)\lambda(dx)\), that is \(p\) and \(q\) are the Radon–Nikodym derivatives of \(P\) and \(Q\) respectively with respect to \(\lambda\).
This definition does not depend on \(\lambda\), that is, the Hellinger distance between \(P\) and \(Q\) does not change if \(\lambda\) 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,
\begin{equation} H^{2}(P,Q) = {\frac{1}{2}}\int_{\mathcal {X}}\left({\sqrt {P(dx)}}-{\sqrt {Q(dx)}}\right)^{2} ~. \end{equation}
To define the Hellinger distance in terms of elementary probability theory, let \(\lambda\) be the Lebesgue measure, so that \(f = \frac{dP}{d\lambda}\) and \(q = \frac{dQ}{d\lambda}\) are simply probability density functions.
The squared Hellinger distance can be then expressed as a standard calculus integral,
\begin{equation} H^{2}(f,g) = {\frac {1}{2}}\int \left({\sqrt {f(x)}}-{\sqrt {g(x)}}\right)^{2}\,dx=1-\int {\sqrt {f(x)g(x)}}\,dx ~, \end{equation}
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),
\begin{equation} 0\leq H(P,Q)\leq 1 ~. \end{equation}
For two discrete probability distributions \(P = (p_{1}, \ldots , p_{k})\) and \(Q = (q_{1},\ldots ,q_{k})\), their Hellinger distance is defined as,
\begin{equation} H(P,Q) = {\frac {1}{\sqrt {2}}}\;{\sqrt {\sum _{i=1}^{k}({\sqrt {p_{i}}}-{\sqrt {q_{i}}})^{2}}} ~, \end{equation}
which is directly related to the Euclidean norm of the difference of the square root vectors,
\begin{equation} H(P,Q) = {\frac {1}{\sqrt {2}}}\;{\bigl \|}{\sqrt {P}}-{\sqrt {Q}}{\bigr \|}_{2} ~. \end{equation}
It follows that,
\begin{equation} 1 - H^{2}(P, Q) = \sum_{i=1}^{k}{\sqrt{p_{i} q_{i}}} ~. \end{equation}
\begin{equation} H(P,Q) = {\sqrt{1 - BC(P,Q)}} ~. \end{equation}
\begin{equation} 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}}}} ~. \end{equation}
\begin{equation} 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\} ~. \end{equation}
\begin{equation} H^{2}(P,Q) = 1 - {\frac {2{\sqrt {\alpha \beta }}}{\alpha +\beta }} ~. \end{equation}
\begin{equation} H^{2}(P, Q) = 1 - {\frac {2(\alpha \beta )^{k/2}}{\alpha ^{k}+\beta ^{k}}} ~. \end{equation}
\begin{equation} H^{2}(P, Q) = 1 - e^{-{\frac {1}{2}}({\sqrt {\alpha }}-{\sqrt {\beta }})^{2}} ~. \end{equation}
\begin{equation} 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})}}} ~, \end{equation}
where \(B\) represents the beta function.\begin{equation} 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})}}} ~, \end{equation}
where \(\Gamma\) is the gamma function.The Hellinger distance \(H(P, Q)\) and the total variation distance (or statistical distance) \(\delta(P,Q)\) are related as follows,
\begin{equation} H^{2}(P, Q)\leq \delta(P, Q)\leq {\sqrt{2}}H(P, Q) ~. \end{equation}
These inequalities follow immediately from the inequalities between the 1-norm and the 2-norm.
Final Remarks ⛓
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For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
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| character(*, SK), parameter pm_distanceHellinger::MODULE_NAME = "@pm_distanceHellinger" |
Definition at line 162 of file pm_distanceHellinger.F90.
1.9.3