  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.
