  In large-scale studies with parallel signal-plus-noise observations, the
local false discovery rate is a summary statistic that is often presumed to be
equal to the posterior probability that the signal is null. We prefer to call
the latter quantity the local null-signal rate to emphasize our view that a
null signal and a false discovery are not identical events. The local
null-signal rate is commonly estimated through empirical Bayes procedures that
build on the `zero density assumption', which attributes the density of
observations near zero entirely to null signals. In this paper, we argue that
this strategy does not furnish estimates of the local null-signal rate, but
instead of a quantity we call the complementary local activity rate (clar).
Although it is likely to be small, an inactive signal is not necessarily zero.
The local activity rate addresses two shortcomings of the local null-signal
rate. First, it is a weakly continuous functional of the signal distribution,
and second, it takes on sensible values when the signal is sparse but not
exactly zero. Our findings clarify the interpretation of local false-discovery
rates estimated under the zero density assumption.
