In this article the "Chernoff distance" between two data sets is defined as:

$$ C D= \frac{1}{2} \alpha_{\mathrm{c}} \alpha_{\mathrm{f}}\left(\mu_{\mathrm{c}}-\mu_{\mathrm{f}}\right)^{ T}\left[\alpha_{\mathrm{c}} \Sigma_{\mathrm{c}}+\alpha_{\mathrm{f}} \Sigma_{\mathrm{f}}\right]^{-1}\left(\mu_{\mathrm{c}}-\mu_{\mathrm{f}}\right)+\frac{1}{2} \log \frac{\left|\alpha_{\mathrm{c}} \Sigma_{\mathrm{c}}+\alpha_{\mathrm{f}} \Sigma_{\mathrm{f}}\right|}{\left|\Sigma_{\mathrm{c}}\right|^{\alpha_{c}}\left|\Sigma_{\mathrm{f}}\right|^{\alpha_{\mathrm{f}}}}$$

where $\alpha$, $\mu$ and $\Sigma$ are the "percentages, means and covariances" of both groups, and the "superscripts $T$ and $−1$ refer to the transpose vector and to the inverse of the matrix, respectively".

The original article referenced as the source of this expression is Chernoff (1952), but I could not find it inside.

I also could not find a proper definition for this distance anywhere else. Is this expression correct? Is it similar to an Anderson-Darling test, or Kullback–Leibler divergence, or Bhattacharyya distance?

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    $\begingroup$ Why is there an integral sign in the exponent? $\endgroup$ – innisfree May 23 at 2:53
  • $\begingroup$ It's a typo in the final version of the article, the arXived version shows the proper equation. I've fixed it. $\endgroup$ – Gabriel May 26 at 20:05

It looks like the Bhattacharya distance for multivariate normal distributions, see wiki, when the factors $\alpha =1/2$.

As that distance is $$ B= \int \sqrt{f g} dx $$ I speculate that the Chernoff distance is $$ \int f^\alpha g^{1-\alpha} dx $$ and that this formulae for multivariate normals would give us your expression.

Furthermore if the covariance matrixes are equal, it reduces to a distance that is equivalent to the Mahalanoblis distance

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  • $\begingroup$ You are right, it does look almost exactly like the Bhattacharyya distance for multivariate normal distributions. But still, where did the authors got this expression as the "Chernoff distance"? $\endgroup$ – Gabriel May 23 at 13:05

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