  There is a significant literature on methods for incorporating knowledge into
multiple testing procedures so as to improve their power and precision. Some
common forms of prior knowledge include (a) beliefs about which hypotheses are
null, modeled by non-uniform prior weights; (b) differing importances of
hypotheses, modeled by differing penalties for false discoveries; (c) multiple
arbitrary partitions of the hypotheses into (possibly overlapping) groups; and
(d) knowledge of independence, positive or arbitrary dependence between
hypotheses or groups, suggesting the use of more aggressive or conservative
procedures. We present a unified algorithmic framework called p-filter for
global null testing and false discovery rate (FDR) control that allows the
scientist to incorporate all four types of prior knowledge (a)-(d)
simultaneously, recovering a variety of known algorithms as special cases.
