  We apply a likelihood analysis to pulsar detections, pulsar upper limits, and
diffuse background measurements from the OSSE and EGRET instruments on the
Compton Gamma Ray Observatory to constrain the luminosity law for gamma-ray
pulsars and some properties of the gamma-ray pulsar population. We find that
the dependence of luminosity on spin period and dipole magnetic field is much
steeper at OSSE than at EGRET energies (50-200 keV and >100 MeV, respectively),
suggesting that different emission mechanisms are responsible for low- and
high-energy gamma-ray emission. Incorporating a spin-down model and assuming a
pulsar spatial distribution, we estimate the fraction of the Galactic gamma-ray
background due to unidentified pulsars and find that pulsars may be an
important component of the OSSE diffuse flux, but are most likely not important
at EGRET energies. Using measurements of the diffuse background flux from these
instruments, we are able to place constraints on the braking index, initial
spin period, and magnetic field of the Galactic pulsar population. We are also
able to constrain the pulsar birthrate to be between 1/(25 yr) and 1/(500 yr).
Our results are based on a large gamma-ray beam, but they do not scale in a
simple way with beam size. With our assumed beam size, the implied gamma-ray
efficiency for the EGRET detections is no more than 20%. We estimate that about
20 of the 169 unidentified EGRET sources are probably gamma-ray pulsars. We use
our model to predict the pulsar population that will be seen by future
gamma-ray instruments and estimate that GLAST will detect roughly 750 gamma-ray
pulsars as steady sources, only 120 of which are currently known radio pulsars.
