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You can do this by generating heatmaps from a random subset (say, 30) of the posterior draws, then determining whether the actual heatmap looks like the estimated one

You can also make a version of the heatmap based on conditional distributions by dividing the cell frequencies by the length of the IU.

• Take the values from the posterior draws $\hat{\theta}_1, \hat{\theta}_2, ..., \hat{\theta}_K$ (say; these are all vectors including all of the parameters)
• For each of those posterior draws, simulate a fake dataset $(\hat{L}, \hat{Pl})$ based on the probability model.
• Get some summary statistics from each simulated dataset (e.g. sample mean, sample variance), and then see whether the empirical sample mean, variance, etc. from the dataset is not too unusual (say, between the 2.5% and 97.5% percentiles of the posteriors).

For the word 'he':

• Take both the Poisson and the negative binomial models we got, and use the maximum a posteriori (MAP) estimates.
• Using the PMFs of the Poisson and negative binomial models, get and plot the probability distribution on a graph, along with the empirical PMF from the word's frequency distribution.
• Which one is closer? If the Poisson is closer than the negative binomial, but still not too close, we might want to try a generalised Poisson too.