# Why is the KL divergence the number of bits required to represent the error of an estimator?

I am familiar with several interpretations of the KL divergence, last week I heard of a new one, mentioned in a lecture on probabilistic graphical models. It was stated kind of offhandedly, so I hope I'm getting the gist, but I seem to remember something like

"The KL divergence between a distribution $$\mathcal{D}$$ and an empirical distribution $$\mathcal{D}_{emp}$$ based on a sample $$\mathcal{X}\sim\mathcal{D}$$ is the number of bits required to represent the error of a MLE which is based on $$\mathcal{X}$$"

(I know this sounds weird, MLE for what parameter? Maybe for any parameter? I'm honestly not sure)

Is anyone familiar with such a result?