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We infer the number of planets-per-star as a function of orbital period andplanet size using $Kepler$ archival data products with updated stellarproperties from the $Gaia$ Data Release 2. Using hierarchical Bayesian modelingand Hamiltonian Monte Carlo, we incorporate planet radius uncertainties into aninhomogeneous Poisson point process model. We demonstrate that this modelcaptures the general features of the outcome of the planet formation andevolution around GK stars, and provides an infrastructure to use the $Kepler$results to constrain analytic planet distribution models. We report anincreased mean and variance in the marginal posterior distributions for thenumber of planets per $GK$ star when including planet radius measurementuncertainties. We estimate the number of planets-per-$GK$ star between 0.75 and2.5 $R_{\oplus}$ and 50 to 300 day orbital periods to have a $68\%$ credibleinterval of $0.49$ to $0.77$ and a posterior mean of $0.63$. This posterior hasa smaller mean and a larger variance than the occurrence rate calculated inthis work and in Burke et al. (2015) for the same parameter space using the$Q1-Q16$ (previous $Kepler$ planet candidate and stellar catalog). We attributethe smaller mean to many of the instrumental false positives at longer orbitalperiods being removed from the $DR25$ catalog. We find that the accuracy andprecision of our hierarchical Bayesian model posterior distributions are lesssensitive to the total number of planets in the sample, and more so on thecharacteristics of the catalog completeness and reliability and the span of theplanet parameter space.

Sensitivity Analyses of Exoplanet Occurrence Rates from Kepler and Gaia