  The accurate spatial location of gamma-ray bursts (GRBs) is crucial for both
producing a detector response matrix (DRM) and follow-up observations by other
instruments. The Fermi Gamma-ray Burst Monitor (GBM) has the largest field of
view (FOV) for detecting GRBs as it views the entire unocculted sky, but as a
non-imaging instrument it relies on the relative count rates observed in each
of its 14 detectors to localize transients. Improving its ability to accurately
locate GRBs and other transients is vital to the paradigm of multi-messenger
astronomy, including the electromagnetic follow-up of gravitational wave
signals. Here we present the BAyesian Location Reconstruction Of GRBs ({\tt
BALROG}) method for localizing and characterising GBM transients. Our approach
eliminates the systematics of previous approaches by simultaneously fitting for
the location and spectrum of a source. It also correctly incorporates the
uncertainties in the location of a transient into the spectral parameters and
produces reliable positional uncertainties for both well-localized sources and
those for which the GBM data cannot effectively constrain the position. While
computationally expensive, {\tt BALROG} can be implemented to enable quick
follow-up of all GBM transient signals. Also, we identify possible response
problems that require attention as well as caution when using standard, public
GBM DRMs. Finally, we examine the effects of including the variance in location
on the spectral parameters of GRB 080916C. We find that spectral parameters
change and no extra components are required when these effects are included in
contrast to when we use a fixed location. This finding has the potential to
alter both the GRB spectral catalogs as well as the reported spectral
composition of some well-known GRBs.
