# 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.

Notes

New 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 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.

References

1

Grant, Boyd, and Ye, “CVX: Matlab Software for Disciplined Convex Programming”, http://cvxr.com/cvx/