  The identification and description of point sources is one of the oldest
problems in astronomy; yet, even today the correct statistical treatment for
point sources remains one of the field's hardest problems. For dim or crowded
sources, likelihood based inference methods are required to estimate the
uncertainty on the characteristics of the source population. In this work, a
new parametric likelihood is constructed for this problem using Compound
Poisson Generator (CPG) functionals which incorporate instrumental effects from
first principles. We demonstrate that the CPG approach exhibits a number of
advantages over Non-Poissonian Template Fitting (NPTF) - an existing method -
in a series of test scenarios in the context of X-ray astronomy. These
demonstrations show that the effect of the point-spread function, effective
area, and choice of point-source spatial distribution cannot, generally, be
factorised as they are in NPTF, while the new CPG construction is validated in
these scenarios. Separately, an examination of the diffuse-flux emission limit
is used to show that most simple choices of priors on the standard
parameterisation of the population model can result in unexpected biases: when
a model comprising both a point-source population and diffuse component is
applied to this limit, nearly all observed flux will be assigned to either the
population or to the diffuse component. A new parametrisation is presented for
these priors which properly estimates the uncertainties in this limit. In this
choice of priors, CPG correctly identifies that the fraction of flux assigned
to the population model cannot be constrained by the data.
