  We propose a framework to analyze and quantify the bias in adaptive data
analysis. It generalizes that proposed by Russo and Zou'15, applying to
measurements whose moment generating function exists, measurements with a
finite $p$-norm, and measurements in general Orlicz spaces. We introduce a new
class of dependence measures which retain key properties of mutual information
while more effectively quantifying the exploration bias for heavy tailed
distributions. We provide examples of cases where our bounds are nearly tight
in situations where the original framework of Russo and Zou'15 does not apply.
