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Plug-in Estimators for Conditional Expectations and Probabilities

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We study plug-in estimators of conditional expectations and probabilities, and we provide a
systematic analysis of their rates of convergence. The plug-in approach is particularly useful in
this setting since it introduces a natural link to VC- and empirical process theory. We make use
of this link to derive rates of convergence that hold uniformly over large classes of functions
and sets, and under various conditions. For instance, we demonstrate that elementary conditional
probabilities are estimated by these plug-in estimators with a rate of n˛1=2 if one conditions
with a VC-class of sets and where ˛ 2 Œ0; 1=2/ controls a lower bound on the size of sets we
can estimate given n samples. We gain similar results for Kolmogorov’s conditional expectation
and probability which generalize the elementary forms of conditioning. Due to their simplicity,
plug-in estimators can be evaluated in linear time and there is no up-front cost for inference

Bibliographic note

Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) 2018, Lanzarote, Spain. PMLR: Volume 84. Copyright 2018 by the author(s).