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> it doesn’t satisfy some basic properties a real distance (a norm) would satisfy, including the fact that it isn’t symmetric [...] and it does not satisfy the triangle inequality.

Not sure about "real" but one can have useful distances which are not symmetric like the distance between cities measured in time or in gallons.

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It just needs to be clarified that KL divergence isn’t a proper mathematical norm, so it doesn’t behave the way we intuitively think a distance should. As mentioned, it doesn’t satisfy the triangle inequality, which is a basic property for any distance-like function.

In comparison, both of your examples are much closer to norms as they both satisfy the triangle inequality.

For reference, this is what I’m referring to when I say a “norm”:

https://en.m.wikipedia.org/wiki/Norm_(mathematics)

I was just clarifying that norm and distance are not the same as you seemed to imply with "a real distance (a norm)". (And I think one can also intuitively understand that the distance from A to B may not be the same as the distance from B to A as soon as we step outside of geometry.)

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