  The vast majority of gravitational-wave signals from stellar-mass compact
binary mergers are too weak to be individually detected with present-day
instruments and instead contribute to a faint, persistent background. This
astrophysical background is targeted by searches that model the
gravitational-wave ensemble collectively with a small set of parameters. The
traditional search models the background as a stochastic field and estimates
its amplitude by cross-correlating data from multiple interferometers. A
different search uses gravitational-wave templates to marginalize over all
individual event parameters and measure the duty cycle and population
properties of binary mergers. Both searches ultimately estimate the total
merger rate of compact binaries and are expected to yield a detection in the
coming years. Given the conceptual and methodological differences between them,
though, it is not well understood how their results should be mutually
interpreted. In this paper, we use the Fisher information to study the
implications of a background detection in terms of which region of the Universe
each approach probes. Specifically, we quantify how information about the
compact binary merger rate is accumulated by each search as a function of the
event redshift. For the LIGO Design sensitivity and a
uniform-in-comoving-volume distribution of equal-mass 30M_sol binaries, the
traditional cross-correlation search obtains 99% of its information from
binaries up to redshift 2.5 (average signal-to-noise-ratio <8), and the
template-based search from binaries up to redshift 1.0 (average
signal-to-noise-ratio ~8). While we do not calculate the total information
accumulated by each search, our analysis emphasizes the need to pair any
claimed detection of the stochastic background with an assessment of which
binaries contribute to said detection.
