# If the KL divergence is not a metric or a measure, what is it?

The KL divergence is not a metric because e.g. it does not satisfy the symmetry property that metrics posses. According to the definition of measure, the KL divergence doesn't seem to be a measure, although the related Wikipedia article introduces the KL divergence as "a measure of how one probability distribution is different from a second". So, mathematically, what is the KL divergence?

Note that I know how the KL divergence is defined, so don't tell me how it's defined.

• It is a measure in the English sense. You can use the word the way the Wikipedia article does. You just have to avoid confusing measure theorists. :) – hobbs May 13 at 0:49
• @hobbs You can, but we should avoid it, given the confusion that it can cause. Also, if it's a pre-metric, why not calling it as such? Just get used to this new terminology, if you were not familiar with it. – nbro May 13 at 2:10

1. $$d(x, y) \geq 0, \forall x, y$$
2. $$d(x, x) = 0, \forall x$$