  We discover a connection between the Benjamini-Hochberg (BH) procedure and
the recently proposed e-BH procedure [Wang and Ramdas, 2022] with a suitably
defined set of e-values. This insight extends to a generalized version of the
BH procedure and the model-free multiple testing procedure in Barber and
Cand\`es [2015] (BC) with a general form of rejection rules. The connection
provides an effective way of developing new multiple testing procedures by
aggregating or assembling e-values resulting from the BH and BC procedures and
their use in different subsets of the data. In particular, we propose new
multiple testing methodologies in three applications, including a hybrid
approach that integrates the BH and BC procedures, a multiple testing procedure
aimed at ensuring a new notion of fairness by controlling both the group-wise
and overall false discovery rates (FDR), and a structure adaptive multiple
testing procedure that can incorporate external covariate information to boost
detection power. One notable feature of the proposed methods is that we use a
data-dependent approach for assigning weights to e-values, significantly
enhancing the efficiency of the resulting e-BH procedure. The construction of
the weights is non-trivial and is motivated by the leave-one-out analysis for
the BH and BC procedures. In theory, we prove that the proposed e-BH procedures
with data-dependent weights in the three applications ensure finite sample FDR
control. Furthermore, we demonstrate the efficiency of the proposed methods
through numerical studies in the three applications.
