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# scipy.special.rel_entr

scipy.special.rel_entr(x, y, out=None) = <ufunc 'rel_entr'>

  Elementwise function for computing relative entropy.

  \[\begin{split}\mathrm{rel\_entr}(x, y) = \begin{cases} x \log(x / y) & x > 0, y > 0 \\ 0 & x = 0, y \ge 0 \\ \infty & \text{otherwise} \end{cases}\end{split}\]
  Parameters:
    x, y array_like

      Input arrays

    out ndarray, optional

      Optional output array for the function results

  Returns:
    scalar or ndarray

      Relative entropy of the inputs

  See also

  `entr`, `kl_div`, `scipy.stats.entropy`

  Notes

  New in version 0.15.0.

  This function is jointly convex in x and y.

  The origin of this function is in convex programming; see [1]. Given two discrete probability distributions \(p_1, \ldots, p_n\) and \(q_1, \ldots, q_n\), the definition of relative entropy in the context of information theory is

  \[\sum_{i = 1}^n \mathrm{rel\_entr}(p_i, q_i).\]

  To compute the latter quantity, use `scipy.stats.entropy`.

  See [2] for details.

  References

  [1]

  Boyd, Stephen and Lieven Vandenberghe. Convex optimization. Cambridge University Press, 2004. DOI:https://doi.org/10.1017/CBO9780511804441

  [2]

  Kullback-Leibler divergence, https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence