  Sufficient statistics are combinations of data in terms of which the
likelihood function can be rewritten without loss of information. Depending on
the data volume reduction, the use of sufficient statistics as a preliminary
step in a Bayesian analysis can lead to significant increases in efficiency
when sampling from posterior distributions of model parameters. Here we show
that the frequency integrand of the cross-correlation statistic and its
variance are approximate sufficient statistics for ground-based searches for
stochastic gravitational-wave backgrounds. The sufficient statistics are
approximate because one works in the weak-signal approximation and uses
measured estimates of the auto-correlated power in each detector. Using
analytic and numerical calculations, we prove that LIGO-Virgo's hybrid
frequentist-Bayesian parameter estimation analysis is equivalent to a fully
Bayesian analysis. This work closes a gap in the LIGO-Virgo literature, and
suggests directions for additional searches.
