In astrophysics, the distributions of fluxes or intensities of populations of sources provide insight into the structure and evolution of the universe. We propose a science-driven hierarchical Bayesian framework to estimate the X-ray flux distribution of a population of galaxies. The distribution is parameterized via a broken power law model, formulated as a mixture of Pareto distributions, and is fit using data from a combination of non-representative X-ray and optical surveys. Because fainter sources are less likely to be detected, astronomical surveys suffer from incompleteness at the low-intensity end, and dedicated survey simulations are used to study the resulting missing-data pattern and determine the form of corrections. We introduce a methodology to combine data from two (or more) surveys with different incompleteness functions. This allows us to leverage deeper surveys (e.g., optical) to more fully characterize the X-ray log(N)-log(S) (where fewer sources are detected). We take advantage of a well-calibrated characterization of the missing-data pattern, particularly in X-ray surveys, to make incompleteness corrections for sources that are not detected in either survey. The proposed model is validated using a simulation study designed to mimic the Chandra Deep Field South (CDF-S), demonstrating accurate recovery of the flux distribution even under high proportions of systematic incompleteness. Finally, we apply our proposed method to data from the CDF-S.

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