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Does anyone know of any analytical expression for the upper bound on the Kullback–Leibler divergence for a discrete random variable?

What I am looking for is the bound expressed as

0 <= S_KL <= f(k)

Where k is the number of distinguishable outcomes.

Ultimately I am also looking for the bound in the case where the probability itself belongs to a special discrete set, rather than R. (This would correspond to combinatorics of microstates instead of a the continuum probability; edit: this means there is another variable entering the bound which is the sample/memory size, but the first question is what the limit is in sample size -> infinity.)

I think I looked for this long time ago. If anyone happens to just know a pointer to where this is worked out I would be interested.

The context of the question is to find a deeper understanding various of the entropy bounds in physics.

I am to start with not sure to what extent there are analytical expressions or if I need to make some computer simulations.

Edit: I'm not looking for some approximate (large k) bounds, I am looking for the actual dependence of the upper bound on k.

/Fredrik

What I am looking for is the bound expressed as

0 <= S_KL <= f(k)

Where k is the number of distinguishable outcomes.

Ultimately I am also looking for the bound in the case where the probability itself belongs to a special discrete set, rather than R. (This would correspond to combinatorics of microstates instead of a the continuum probability; edit: this means there is another variable entering the bound which is the sample/memory size, but the first question is what the limit is in sample size -> infinity.)

I think I looked for this long time ago. If anyone happens to just know a pointer to where this is worked out I would be interested.

The context of the question is to find a deeper understanding various of the entropy bounds in physics.

I am to start with not sure to what extent there are analytical expressions or if I need to make some computer simulations.

Edit: I'm not looking for some approximate (large k) bounds, I am looking for the actual dependence of the upper bound on k.

/Fredrik

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