  We characterize the eccentricity distribution of a sample of ~50 short-period
planet candidates using transit and occultation measurements from NASA's Kepler
Mission. First, we evaluate the sensitivity of our hierarchical Bayesian
modeling and test its robustness to model misspecification using simulated
data. When analyzing actual data assuming a Rayleigh distribution for
eccentricity, we find that the posterior mode for the dispersion parameter is
$\sigma=0.081 \pm^{0.014}_{0.003}$. We find that a two-component Gaussian
mixture model for $e \cos \omega$ and $e \sin \omega$ provides a better model
than either a Rayleigh or Beta distribution. Based on our favored model, we
find that $\sim90\%$ of planet candidates in our sample come from a population
with an eccentricity distribution characterized by a small dispersion
($\sim0.01$), and $\sim10\%$ come from a population with a larger dispersion
($\sim0.22$). Finally, we investigate how the eccentricity distribution
correlates with selected planet and host star parameters. We find evidence that
suggests systems around higher metallicity stars and planet candidates with
smaller radii come from a more complex eccentricity distribution.
