Detection of point sources in images is a fundamental operation in astrophysics, and is crucial for constraining population models of the underlying point sources or characterizing the background emission. Standard techniques fall short in the crowded-field limit, losing sensitivity to faint sources and failing to track their covariance with close neighbors. We construct a Bayesian framework to perform inference of faint or overlapping point sources. The method involves probabilistic cataloging, where samples are taken from the posterior probability distribution of catalogs consistent with an observed photon count map. In order to validate our method we sample random catalogs of the gamma-ray sky in the direction of the North Galactic Pole (NGP) by binning the data in energy and Point Spread Function (PSF) classes. Using three energy bins spanning $0.3 - 1$, $1 - 3$ and $3 - 10$ GeV, we identify $270substack{+30 -10}$ point sources inside a $40^circ times 40^circ$ region around the NGP above our point-source inclusion limit of $3 times 10^{-11}$/cm$^2$/s/sr/GeV at the $1-3$ GeV energy bin. Modeling the flux distribution as a power law, we infer the slope to be $-1.92substack{+0.07 -0.05}$ and estimate the contribution of point sources to the total emission as $18substack{+2 -2}$%. These uncertainties in the flux distribution are fully marginalized over the number as well as the spatial and spectral properties of the unresolved point sources. This marginalization allows a robust test of whether the apparently isotropic emission in an image is due to unresolved point sources or of truly diffuse origin.
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