Extending the Hellinger Distance of discrete probability distributions to multivariate distributions

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?

• Every bivariate discrete distribution can be treated as univariate. Jan 7, 2015 at 8:49
• treat as univariate? you mean, change them into a vector, pij-> pk, qij->qk ? Jan 7, 2015 at 9:13
• $P(X=x_{i},Y=y_{j})=p_{ij} \equiv P(Z=k)=p_k$, where $k$ is the number of combination $ij$. Jan 7, 2015 at 9:28
• 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.^.^ Jan 7, 2015 at 14:50
• stats.stackexchange.com/questions/132562/… Jan 7, 2015 at 14:56