A Probabilistic Framework for Population Studies of the Solar Neighborhood: Application to SDSS-V and Gaia

Studies of the Solar Neighborhood require spectroscopic follow-up of stars identified in astrometric surveys to fully characterize their physical properties. The SDSS-V Solar Neighborhood Census (SNC) is a dedicated program to observe stars within 100 pc. However, due to competing observing programs and fiber assignment constraints, the resulting sample carries severe and complex selection effects. A framework is presented for characterizing the selection function of the SDSS-V SNC relative to the Gaia Catalog of Nearby Stars (GCNS), along with a forward modeling method to infer the properties of stellar subpopulations across the GCNS-defined 100 pc sample. The selection function is based on a method that models the selection probability as a function of sky position, Gaia G magnitude, and BP-RP. The resulting detection probabilities faithfully reproduce the known survey planning logic. This work further introduces the concept of a "subpopulation probability" -- a grid of posterior estimates across the Hertzsprung-Russell (HR) diagram representing the likelihood that a GCNS member belongs to a given SDSS-V defined subpopulation. The framework is validated with a mock dataset and its scientific utility is demonstrated through two applications using data from the Data Release 19: mapping H alpha emission across the HR diagram and measuring the variation of stellar density with mass and metallicity. These results illustrate how statistically robust population studies can be conducted with an incomplete spectroscopic survey when the selection function is well characterized. The code is made publicly available with this work, which will serve as an important tool for future studies.