  To test whether the short GRB rates, redshift distribution and host galaxies
are consistent with current theoretical predictions, we use avery large
database of population synthesis calculations to examine BH-NS and NS-NS merger
rates in the universe, factoring in (i) the star formation history of the
universe, (ii) a heterogeneous population of star-forming galaxies, including
spirals and ellipticals, and (iii) a simple flux-limited selection model for
short GRB detection. When we require our models reproduce the known short GRB
rates and redshift measurements (and, for NS-NS, the merger rates extrapolated
from binary pulsars in the Galaxy), a small fraction of models reproduce all
observations, both when we assume a NS-NS and a BH-NS origin for bursts. Most
commonly models produce mergers preferentially in spiral galaxies if short GRBs
arise from NS-NS mergers alone. Model universes where present-day binary
mergers occur preferentially in elliptical galaxies necessarily include a
significant fraction of binaries with long delay times between birth and merger
(often $O(10{\rm Gyr})$). Though long delays occur, almost all of our models
predict that a higher proportion of short GRBs should occur at moderate to high
redshift (e.g., $z>1$) than has presently been observed, in agreement with
recent observations which suggest a selection bias towards successful follow-up
of low-redshift short GRBs. Finally, if only a fraction of BH-NS mergers have
the right combination of masses and spins to make GRBs, then at best only a
small fraction of BH-NS models could be consistent with all {\em current}
available data. (Abridged)
