  We consider an extension of $\epsilon$-entropy to a KL-divergence based
complexity measure for randomized density estimation methods. Based on this
extension, we develop a general information-theoretical inequality that
measures the statistical complexity of some deterministic and randomized
density estimators. Consequences of the new inequality will be presented. In
particular, we show that this technique can lead to improvements of some
classical results concerning the convergence of minimum description length and
Bayesian posterior distributions. Moreover, we are able to derive clean
finite-sample convergence bounds that are not obtainable using previous
approaches.
