  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 as 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
advantages over Non-Poissonian Template Fitting (NPTF) - an existing parametric
likelihood 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, in general, be factorised as they are in the NPTF construction, 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 is demonstrated to
properly estimate the uncertainties in this limit. In this choice of priors,
the CPG correctly identifies that the fraction of flux assigned to the
population model cannot be constrained by the data.
