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Statistical Risk Characterization of Penalized Likelihood Procedures: An Information-Theoretic Determination

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We review theory for the Minimum Description Length principle, penalized likelihood and its statistical risk. An information theoretic condition on a penalty pen(f) yields the conclusion that the optimizer of the penalized log likelihood criterion log 1/likelihood(f) + pen(f) has risk not more than the index of resolvability, corresponding to the accuracy of the optimizer of the expected value of the criterion. For the linear span of a dictionary of candidate terms, we develop the information theoretic validity of penalties based on the l_1 norm of the coefficients in regression and log-density estimation settings. New results are presented for Gaussian graphical models. This represents joint work with Xi Luo and Sabyasachi Chatterjee.

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