tokens: not directly calculated. Instead, Equations ( 6 ) and ( 7 ) are combined by using the posterior samples: to define the likelihood that one then draws from:Performing these two steps repeatedly produces an ensemble of data sets drawn from Equation ( 6 ). Applying Equation ( 8 ) to our M–R relation requires evaluating the lower levels of the forward model described in Equation ( 4 ). Part of the subtlety of checking data-model consistency arises from the hierarchical nature of our model. In particular, of Equation ( 8 ) includes both the population-wide parameters C , γ , σ M and the individual parameters we can use the posterior samples of either of these groups of parameters to calculate . We show the result of both choices in Figure 5 , using the posterior samples from the baseline M–R relation (second line of Table 1 ). Conceptually, using samples from the individual true mass and radius posteriors () evaluates the fit of the model to the currently observed set of planets (the green histograms on the left), while using posterior samples for the M–R relation parameters () evaluates the fit of the model to altogether new sets of planets (the blue histograms on the right). For the former case, the same and as the observed data set are used; for the latter case, is drawn from the distribution of for the observed planets that have a similar mass, and is drawn from the observed data set’s full distribution of without controlling for radius (the size of the radius error bars are fairly constant across the data set). The second aspect of hierarchical model-checking which requires some effort is the identification of a robust statistic to quantify the discrepancy between the observed data and the model-generated data. Optimally this statistic would test the fit of every part of the modeled probability distribution, including the “average” behavior of the model, the “extreme” behavior out on the tails of the distribution, and for hierarchical models, the accuracy of the grouping and relational structure that is illustrated by graphs like Figure 1 . Due to high dimensional parameter space, this proves to be very difficult, so one must identify several statistics which test such aspects separately. To illustrate the problem, we choose two: f 1 σ , the fraction of a given data set’s simulated mass, radius observations which fall within the 68% coverage interval of our probabilistic, baseline M–R relation (blue region in Figure 4 ), and f μ , the fraction of a given data set’s mass measurements whose 1 σ error bars cross the mean relation μ (see Equation ( 2 )) of that same model. Therefore, f 1 σ tests how well the width of our probabilistic M–R relation fits the data, and f μ tests how tightly grouped the data are around the mean compared to the normal distribution of our model. The results of these tests are displayed in Figure 5 . The observed data’s f 1 σ statistic (top) falls at the 63rd and 83rd percentiles (i.e., within “one sigma”) of the
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