We construct a Bayesian framework to perform inference of dim 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. By implementing across-model jumps between point source models of different dimensionality, we collect a representative ensemble of catalogs consistent with the image. 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 PSF (Point Spread Function) classes. Using three energy bins between $0.3 - 1$, $1 - 3$ and $3 - 10$ GeV, we identify $270\substack{+30 -10}$ 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. Most of these point sources are time-variable blazars. Modeling the flux distribution as a single power law, we infer the slope to be $-1.92\substack{+0.07 -0.05}$ and estimate the contribution of point sources (resolved and unresolved) to the total emission as $18\substack{+2 -2}$\%. Further analyses that rely on the ensemble of sample catalogs instead of only the most likely catalog, can perform reliable marginalization over uncertainties in the number as well as spatial and spectral properties of the 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. With the increase in the availability of computational resources in the near future, probabilistic cataloging can potentially be applied to full sky datasets or optical images and replace the standard data reduction pipelines for crowded fields.
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