We present constraints on an astrophysical population of neutrino sources imposed by recent data from the IceCube neutrino observatory. By using the IceCube point source search method to model the detection of sources, our detection criterion is more sensitive than using the observation of high-energy neutrino multiplets for source identification. We frame the problem as a Bayesian hierarchical model to connect the high-level population parameters to the IceCube data, allowing us to consistently account for all relevant sources of uncertainty in our model assumptions. Our results show that sources with a local density of
n
0
≳
10
−
7
 
 
Mpc
−
3
and luminosity
L
≲
10
43
 
 
erg
 
s
−
1
are the most likely candidates, but that populations of rare sources with
n
0
≃
10
−
9
 
 
Mpc
−
3
and
L
≃
10
45
 
 
erg
 
s
−
1
can still be consistent with the IceCube observations. We demonstrate that these conclusions are strongly dependent on the source evolution considered, for which we consider a wide range of models. In doing so, we present realistic, model-independent constraints on the population parameters that reflect our current state of knowledge from astrophysical neutrino observations. We also use our framework to investigate constraints in the case of possible source detections and future instrument upgrades. Our approach is flexible and can be used to model specific source cases and extended to include multimessenger information.