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Novel Change of Measure Inequalities with Applications to PAC-Bayesian Bounds and Monte Carlo Estimation

Abstract
We introduce several novel change of measure inequalities for two families of divergences: -divergences and -divergences. We show how the variational representation for -divergences leads to novel change of measure inequalities. We also present a multiplicative change of measure inequality for -divergences and a generalized version of Hammersley-Chapman-Robbins inequality. Finally, we present several applications of our change of measure inequalities, including PAC-Bayesian bounds for various classes of losses and non-asymptotic intervals for Monte Carlo estimates.
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