> As a side note, KL divergence is actually symmetric to the second order: This is backwards. Its a tautology. Any function no matter how egregiously asymmetr

> As a side note, KL divergence is actually symmetric to the second order: This is backwards. Its a tautology. Any function no matter how egregiously asymmetric is locally symmetric if it is twice differentiable at that point. This is so by construction, you are approximating it locally by the best possible quadratic [hence locally symmetric] curve [surface]. Hence, the claim about symmetry is not false, but it is vacuous. Much like the claim that the equation of French curve is such that no matter how you turn it at its highest point it makes a tangent with the horizontal. That said Fisher information metric does have many uses.