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For two discrete probability distributions P=(p1..pk) and Q=(q1...qk), their Hellinger distance is defined as

$$H(P,Q)=\frac{1}{\sqrt{2}}\sqrt{\sum_{i=1}^k(\sqrt{p_i}-\sqrt{q_i})^2}$$

could this be extended into bivariate

$$H(P,Q)=\frac{1}{\sqrt{2}}\sqrt{\sum_{i,j}(\sqrt{p_{ij}}-\sqrt{q_{ij}})^2}$$

If this is wrong, is there any other distance metric to measure the distance of such multivariate probability distribution?

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    $\begingroup$ Every bivariate discrete distribution can be treated as univariate. $\endgroup$
    – mpiktas
    Commented Jan 7, 2015 at 8:49
  • $\begingroup$ treat as univariate? you mean, change them into a vector, pij-> pk, qij->qk ? $\endgroup$
    – Jimmy Kang
    Commented Jan 7, 2015 at 9:13
  • $\begingroup$ $P(X=x_{i},Y=y_{j})=p_{ij} \equiv P(Z=k)=p_k$, where $k$ is the number of combination $ij$. $\endgroup$
    – mpiktas
    Commented Jan 7, 2015 at 9:28
  • $\begingroup$ Thank you very much, but I'm still feel confused in my real application, could you read my question on this site, and give me some suggestion.^.^ $\endgroup$
    – Jimmy Kang
    Commented Jan 7, 2015 at 14:50
  • $\begingroup$ stats.stackexchange.com/questions/132562/… $\endgroup$
    – Jimmy Kang
    Commented Jan 7, 2015 at 14:56

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The definition of the Hellinger distance do not depend on dimension, it is the same in every dimension. So the answer to your question is YES, you can compute with a double sum as you propose.

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