  The radius-period distribution of exoplanets has been characterized by the
\textit{Kepler} survey, and the empirical mass-radius relation by the subset of
\textit{Kepler} planets with mass measurements. We combine the two in order to
constrain the joint mass-radius-period distribution of \textit{Kepler}
transiting planets. We employ hierarchical Bayesian modeling and mixture models
to formulate four models with varying complexity and fit these models to the
data. We find that the most complex models that treat planets with significant
gaseous envelopes, evaporated core planets, and intrinsically rocky planets as
three separate populations are preferred by the data and provide the best fit
to the observed distribution of \textit{Kepler} planets. We use these models to
calculate occurrence rates of planets in different regimes and to predict
masses of \textit{Kepler} planets, revealing the model dependent nature of
both. When using models with envelope mass loss to calculate $\eta_\oplus$, we
find nearly an order of magnitude drop, indicating that many Earth-like planets
discovered with \textit{Kepler} may be evaporated cores which do not
extrapolate out to higher orbital periods. This work provides a framework for
higher-dimensional studies of planet occurrence and for using mixture models to
incorporate different theoretical populations of planets.
