From $ε$-entropy to KL-entropy: Analysis of minimum information complexity density estimation

Details From $ε$-entropy to KL-entropy: Analysis of minimum information complexity density estimation From $ε$-entropy to KL-entropy: Analysis of minimum information complexity density estimation

2007-02-22

arxiv.org/abs/math/0702653v1

Annals of Statistics 2006, Vol. 34, No. 5, 2180-2210

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053606000000704 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053606000000704

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.

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