Novel Change of Measure Inequalities with Applications to PAC-Bayesian Bounds and Monte Carlo Estimation
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:1711-1719, 2021.
We introduce several novel change of measure inequalities for two families of divergences: $f$-divergences and $\alpha$-divergences. We show how the variational representation for $f$-divergences leads to novel change of measure inequalities. We also present a multiplicative change of measure inequality for $\alpha$-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.