{\boldsymbol{Z}}}}{P_W}$, which they call *disintegrated* mutual information, in place of the mutual information leads to further improvements on the basic bound.

Recent studies, starting with the work of Steinke and Zakynthinou [@steinke20-a], have considered a setting with more structure, where it is assumed that a set ${\widetilde{\boldsymbol{Z}}}$ consisting of $2n$ independent and identically distributed () training samples from $P_Z$ is available, and that ${\boldsymbol{Z}}$ is formed by selecting $n$ entries of ${\widetilde{\boldsymbol{Z}}}$ at random. We will refer to this setup as the *random-subset setting*, and call the setting without this additional structure the *standard setting*. In the random-subset setting, the average generalization error can be bounded by a quantity that depends on the conditional mutual information between the output hypothesis and the random variable that determines the selected training data ${\boldsymbol{Z}}$, given ${\widetilde{\boldsymbol{Z}}}$ [@steinke20-a Thm. 5.1]. One advantage of this bound over the standard mutual information bound is that the conditional mutual information is always bounded. This broadens the applicability of the bound and results in tighter estimates. Also, as discussed in [@steinke20-a Sec. 4], the conditional mutual information that appears in the bound has strong connections to classical generalization measures, such as VC dimension, compressibility, and stability. In [@Haghifam2020], 