  The field of exoplanet atmospheric characterization is trending towards
comparative studies involving many planetary systems, and using Bayesian
hierarchical modelling is a natural next step. Here we demonstrate two use
cases. We first use hierarchical modelling to quantify variability in repeated
observations by reanalyzing a suite of ten Spitzer secondary eclipse
observations of the hot Jupiter XO-3b. We compare three models: one where we
fit ten separate eclipse depths, one where we use a single eclipse depth for
all ten observations, and a hierarchical model. By comparing the Widely
Applicable Information Criterion of each model, we show that the hierarchical
model is preferred over the others. The hierarchical model yields less scatter
across the suite of eclipse depths -- and higher precision on the individual
eclipse depths -- than does fitting the observations separately. We find that
the hierarchical eclipse depth uncertainty is larger than the uncertainties on
the individual eclipse depths, which suggests either slight astrophysical
variability or that single eclipse observations underestimate the true eclipse
depth uncertainty. Finally, we fit a suite of published dayside brightness
measurements for 37 planets using a hierarchical model of brightness
temperature vs irradiation temperature. The hierarchical model gives tighter
constraints on the individual brightness temperatures than the non-hierarchical
model. Although we tested hierarchical modelling on Spitzer eclipse data of hot
Jupiters, it is applicable to observations of smaller planets like hot neptunes
and super earths, as well as for photometric and spectroscopic transit or phase
curve observations.
