  The direct observation of gravitational waves with Advanced LIGO and Advanced
Virgo offers novel opportunities to test general relativity in strong-field,
highly dynamical regimes. One such opportunity is the measurement of
gravitational-wave polarizations. While general relativity predicts only two
tensor gravitational-wave polarizations, general metric theories of gravity
allow for up to four additional vector and scalar modes. The detection of these
alternative polarizations would represent a clear violation of general
relativity. The LIGO-Virgo detection of the binary black hole merger GW170814
has recently offered the first direct constraints on the polarization of
gravitational waves. The current generation of ground-based detectors, however,
is limited in its ability to sensitively determine the polarization content of
transient gravitational-wave signals. Observation of the stochastic
gravitational-wave background, in contrast, offers a means of directly
measuring generic gravitational-wave polarizations. The stochastic background,
arising from the superposition of many individually unresolvable
gravitational-wave signals, may be detectable by Advanced LIGO at
design-sensitivity. In this paper, we present a Bayesian method with which to
detect and characterize the polarization of the stochastic background. We
explore prospects for estimating parameters of the background, and quantify the
limits that Advanced LIGO can place on vector and scalar polarizations in the
absence of a detection. Finally, we investigate how the introduction of new
terrestrial detectors like Advanced Virgo aid in our ability to detect or
constrain alternative polarizations in the stochastic background. We find that,
although the addition of Advanced Virgo does not notably improve detection
prospects, it may dramatically improve our ability to estimate the parameters
of backgrounds of mixed polarization.
