  We discuss the detection of gravitational-wave backgrounds in the context of
Bayesian inference and suggest a practical definition of what it means for a
signal to be considered stochastic---namely, that the Bayesian evidence favors
a stochastic signal model over a deterministic signal model. A signal can
further be classified as Gaussian-stochastic if a Gaussian signal model is
favored. In our analysis we use Bayesian model selection to choose between
several signal and noise models for simulated data consisting of uncorrelated
Gaussian detector noise plus a superposition of sinusoidal signals from an
astrophysical population of gravitational-wave sources. For simplicity, we
consider co-located and co-aligned detectors with white detector noise, but the
method can be extended to more realistic detector configurations and power
spectra. The general trend we observe is that a deterministic model is favored
for small source numbers, a non-Gaussian stochastic model is preferred for
intermediate source numbers, and a Gaussian stochastic model is preferred for
large source numbers. However, there is very large variation between individual
signal realizations, leading to fuzzy boundaries between the three regimes. We
find that a hybrid, trans-dimensional model comprised of a deterministic signal
model for individual bright sources and a Gaussian-stochastic signal model for
the remaining confusion background outperforms all other models in most
instances.
