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The MDL two-part coding $ \textit{index of resolvability} $ provides afinite-sample upper bound on the statistical risk of penalized likelihoodestimators over countable models. However, the bound does not apply tounpenalized maximum likelihood estimation or procedures with exceedingly smallpenalties. In this paper, we point out a more general inequality that holds forarbitrary penalties. In addition, this approach makes it possible to deriveexact risk bounds of order $1/n$ for iid parametric models, which improves onthe order $(\log n)/n$ resolvability bounds. We conclude by discussingimplications for adaptive estimation.

Finite-sample risk bounds for maximum likelihood estimation with arbitrary penalties