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We introduce a method to infer the vertical distribution of stars in theMilky Way using a Poisson likelihood function, with a view to applying ourmethod to the Gaia catalogue. We show how to account for the sample selectionfunction and for parallax measurement uncertainties. Our method is validatedagainst a simulated sample drawn from a model with two exponential discs and apower-law halo profile. A mock Gaia sample is generated using the Gaiaastrometry selection function, whilst realistic parallax uncertainties aredrawn from the Gaia Astrometric Spread Function. The model is fit to the mockin order to rediscover the input parameters used to generate the sample. Werecover posterior distributions which accurately fit the input parameters towithin statistical uncertainties, demonstrating the efficacy of our method.Using the GUMS synthetic Milky Way catalogue we find that our halo parameterfits can be heavily biased by our overly simplistic model, however, the fits tothe thin and thick discs are not significantly impacted. We apply this methodto Gaia Early Data Release 3 in a companion paper where we also quantify thesystematic uncertainties introduced by oversimplifications in our model.

The Photo-Astrometric Vertical Tracer Density of the Milky Way I: The Method