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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, yarray_like

      Input arrays

    outndarray, 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

  Added 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.

  `rel_entr` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

  Library | CPU | GPU
  --- | --- | ---
  NumPy | ✅ | n/a
  CuPy | n/a | ✅
  PyTorch | ✅ | ✅
  JAX | ✅ | ✅
  Dask | ✅ | n/a

  See Support for the array API standard for more information.

  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