scipy.special.kl_div

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

Elementwise function for computing Kullback-Leibler divergence.

\[\begin{split}\mathrm{kl\_div}(x, y) = \begin{cases} x \log(x / y) - x + y & x > 0, y > 0 \\ y & x = 0, y \ge 0 \\ \infty & \text{otherwise} \end{cases}\end{split}\]

Parameters:

x, yarray_like

Real arguments

outndarray, optional

Optional output array for the function results

Returns:

scalar or ndarray

Values of the Kullback-Liebler divergence.

See also

entr, rel_entr, scipy.stats.entropy

Notes

Added in version 0.15.0.

This function is non-negative and is jointly convex in x and y.

The origin of this function is in convex programming; see [1] for details. This is why the function contains the extra \(-x + y\) terms over what might be expected from the Kullback-Leibler divergence. For a version of the function without the extra terms, see rel_entr.

Array API Standard Support

kl_div 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