We infer the number of planets-per-star as a function of orbital period and planet size using $Kepler$ archival data products with updated stellar properties from the $Gaia$ Data Release 2. Using hierarchical Bayesian modeling and Hamiltonian Monte Carlo, we incorporate planet radius uncertainties into an inhomogeneous Poisson point process model. We demonstrate that this model captures the general features of the outcome of the planet formation and evolution around GK stars, and provides an infrastructure to use the $Kepler$ results to constrain analytic planet distribution models. We report an increased mean and variance in the marginal posterior distributions for the number of planets per $GK$ star when including planet radius measurement uncertainties. We estimate the number of planets-per-$GK$ star between 0.75 and 2.5 $R_{oplus}$ and 50 to 300 day orbital periods to have a $68%$ credible interval of $0.49$ to $0.77$ and a posterior mean of $0.63$. This posterior has a smaller mean and a larger variance than the occurrence rate calculated in this work and in Burke et al. (2015) for the same parameter space using the $Q1-Q16$ (previous $Kepler$ planet candidate and stellar catalog). We attribute the smaller mean to many of the instrumental false positives at longer orbital periods being removed from the $DR25$ catalog. We find that the accuracy and precision of our hierarchical Bayesian model posterior distributions are less sensitive to the total number of planets in the sample, and more so on the characteristics of the catalog completeness and reliability and the span of the planet parameter space.
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