We present a method for multiple hypothesis testing that maintains
control of the False Discovery Rate while incorporating prior
information about the hypothesis. The prior information takes the form
of p-value weights. If the assignment of weights is positively
associated with the null hypotheses being false, the procedure
improves power, except in cases where power is already near one. Even
if the assignment of weights is poor, power is only reduced slightly,
as long as the weights are not too large. We also provide a similar
method to control False Discovery Exceedance.