  We present a new framework to characterize the occurrence rates of planet
candidates identified by Kepler based on hierarchical Bayesian modeling,
Approximate Bayesian Computing (ABC), and sequential importance sampling. For
this study we adopt a simple 2-D grid in planet radius and orbital period as
our model and apply our algorithm to estimate occurrence rates for Q1-Q16
planet candidates orbiting around solar-type stars. We arrive at significantly
increased planet occurrence rates for small planet candidates ($R_p<1.25
R_{\oplus}$) at larger orbital periods ($P>80$d) compared to the rates
estimated by the more common inverse detection efficiency method. Our improved
methodology estimates that the occurrence rate density of small planet
candidates in the habitable zone of solar-type stars is $1.6^{+1.2}_{-0.5}$ per
factor of 2 in planet radius and orbital period. Additionally, we observe a
local minimum in the occurrence rate for strong planet candidates marginalized
over orbital period between 1.5 and 2$R_{\oplus}$ that is consistent with
previous studies. For future improvements, the forward modeling approach of ABC
is ideally suited to incorporating multiple populations, such as planets,
astrophysical false positives and pipeline false alarms, to provide accurate
planet occurrence rates and uncertainties. Furthermore, ABC provides a
practical statistical framework for answering complex questions (e.g.,
frequency of different planetary architectures) and providing sound
uncertainties, even in the face of complex selection effects, observational
biases, and follow-up strategies. In summary, ABC offers a powerful tool for
accurately characterizing a wide variety of astrophysical populations.
